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Текст
WILLIAM FULLER BROWN, JR.
Micromagnetics
Number 18
Interscience Tracts on Physics and Astronomy
INTERSCIENCE TRACTS ON PHYSICS AND ASTRONOMY
Edited by R. E. Marshak
Univcrxily of Rochester
1. D. J. Hughes
NEUTRON OPTICS
2. AI. S. Livingston
HIGH-ENERG1r A CCELERA TORS
3. L. Spitzer, Jr.
PHYSICS OF FULLY IONIZED GASES, secund edition
4. T. G. Cowling
MAGNETOH YDROD YNA MICS
5. D. ter Haar
INTRODUCTION TO THE PHYSICS OF MANY-BODY
SYSTEMS
6. E. J. Opik
PHYSICS OF METEOR FLIGHT IN THE ATMOSPHERE
7. K. Alendelssohn
CRYOPHYSICS
8. J. L. Delcroix
INTRODUCTION TO THE THEORY OF IONIZED GASES
9. T. E. Sterne
AAr INTRODUCTION TO CELESTIAL MECHANICS
10. J. Weber
GENERAL RELATIYITY AND GRAYITATIONAL WAYES
11. R. E. Marshak and E. C. G. Sudarshan
INTRODUCTION TO ELEMENTARY PARTICLE PHYSICS
12. J. L. Olsen
ELECTRON TRANSPORT IN METALS
13. Al. Francon
MODERN APPLICATIONS OF PHYSICAL OPTICS
11. P. B. Jones
THE OPTICAL MODEL IN NUCLEAR AND PARTICLE
PH YSICS
15. R. K. Adair and E. C. Fowler
STRANGE PARTICLES
16. R. Wilson
THE NUCLEON-NUCLEON INTERACTION:
EXPERIMENTAL AND PHENOMENOLOGICAL ASPECTS
17. J. F. Denisse and J. L. Delcroix
PLASMA IFAVKS
18. W. F. Brown, Jr.
MICROMAGNETICS
Additional volumes in preparation
MICROMAGNETICS
WILLIAM FULLER BROWN, JR.
Department of Electrical Engineering
University of Minnesota, Minneapolis, Minnesota
1963 f
INTERSCIENCE PUBLISHERS
a division of John Wiley & Sons, New York • London
Copyright © 1963 by John Wiley & Sons, Inc.
All Rights Reserved
Library of Congress Catalog Card Number 63-18566
Printed in the United States of America
INDIANA UNIVERSITY LIBRARY
Preface
To understand ferromagnetic materials, we must examine them
on a smaller scale than that of ordinary observations. On one such
scale we speak of domains; on another, of lattice sites. This tract
analyzes them on an intermediate scale: small enough to reveal
details of the transition regions between domains, yet large enough
to permit the use of a continuous magnetization vector rather
than of individual atomic spins. , '
This approach to ferromagnetic theory—“micromagnetics”—
began with the well known “wall” calculation of Landau -and
Lifshitz in 1935. Since then there have been various contributions
to it, but mostly as incidental parts of scattered and uncorrelated
investigations. Only since the emergence of rigorous nucleation-
field theory in 1957 has micromagnetics, as such, received conscious
attention.
The object of this tract is to review the origins and underlying
principles of the theory and to present it in a unified way. Dynamic
theory is discussed only enough to show its relation to static
problems. The point of view adopted is phenomenological r atomic
concepts are used only to derive plausible mathematical forms for
thermodynamic or dynamic expressions; discussions that begin in
quantum mechanics and end in micromagnetics seem to me to
exaggerate the dependence of the latter on the former.
Parts of this book, especially Chapter 7, contain results of my
own not previously published in detail; this work was assisted by a
grant from the National Science Foundation. The final work on the
manuscript was completed at the Weizmann Institute of Science,
Rehovot, Israel, where I spent the spring and summer of 1962 as a
Fulbright scholar. I am grateful to the Institute for its hospitality
and to Professor E. H. Frei, of the Electronics Department, for
suggesting the visit and doing much to make it rewarding. For
helpful discussions of many topics I am deeply indebted to Drs. A.
Aharoni, S. Shtrikman, and D. Treves of the Weizmann Institute;
to Dr. E. P. Wohlfarth of Imperial College, London; and to my
Minnesota colleagues and associates Dr. A. H. Morrish, Dr. R. P.
Halverson, and Mr. С. E. Johnson, Jr. Invaluable factors through-
out the preparation of this manuscript have been the encouragement
and help of my wife, Nancy.
William Fuller Brown, Jr.
Department of Electrical Engineering
University of Minnesota
Minneapolis, Minnesota
Juneу 1963
Contents
1. Introduction..................................................... 1
1.1 The General Problem....................................... 1
1.2 Forces Involved........................................... 3
1.3 Methods of Solution....................................... 4
1.4 Methods of Evaluating Energy Functions.................... 9
1.5 Scope of This Book....................................... 10
2. Historical Background........................................... 11
2.1 Early Concepts: Rotation and Inversion................... 11
2.2 Primitive Domain Theory.................................. 12
2.3 The Wall Concept......................................... 15
2.4 Early Calculations in Micromagnetics..................... 16
2.5 Development of Modern Domain Theory...................... 17
2.6 Single-Domain Particles.................................. 19
2.7 Recent Calculations in Micromagnetics.................... 21
3. Principles and Methods ......................................... 24
3.1 Torque Equations, Static and Dynamic..................... 24
3.2 Thermostatic Minimization Principles..................... 26
3.3 Dynamic Variational Principles........................... 30
3.4 Free-Energy Formulas..................................... 32
3.5 Magnetostatic Theorems................................... 38
3.6 Variational Techniques................................... 41
3.7 Treatment of Dissipation................................. 43
4. Basic Static and Dynamic Equations.............................. 46
4.1 Equilibrium Torque Equation.............................. 46
4.2 Static Stability Conditions.............................. 49
4.3 Dynamic Torque Equation.................................. 51
4.4 Steady-State and Transient Problems...................... 53
4.5 Dynamic Criterion for Static Stability................... 54
4.6 Linear Approximation..................................... 55
5. Static Problems: Linearizable Cases............................. 61
5.1 Approach to Saturation................................... 61
5.2 The Coercivity Paradox; the Role of Imperfections .... 66
5.3 General Theory of the Nucleation Field................... 72
5.4 Nucleation-Field Calculation in Specific Cases........... 75
5.5 Approximate Nucleation-Field Calculations................ 82
vn
CONTENTS
• • •
Vlll
6. Static Problems: Nonlinear Calculations....................... 85
6.1 The One-Dimensional Case............................... 85
6.2 The Interdomain Wall................................... 91
6.3 Other One-Dimensional Calculations..................... 94
6.4 The Infinite Cylinder.................................. 96
_ J5.5 Approximate Methods.................................... 98
7. Dynamic Problems...............................................101
7.1 The General Problem....................................101
7.2 Special Cases..........................................102
a. The Uniform Mode...................................102
b. Magnetostatic Modes................................106
c. Periodic Boundary Conditions; Spin Waves...........106
7.3 Resonance Fields and Nucleation Fields.................107
7.4 General Theorems.......................................108
7.5 Example: the Infinite Cylinder.........................112
7.6 Transient Problems.....................................114
8. Magnetostriction...............................................116
8.D The Voigt Approximation................................116
8.2 Free-Energy Formulas...................................120
8.3 Static Effects of Magnetostriction.....................123
8.4 Effect on Nucleation...................................126
8.5 Criticism of the Voigt Approximation...................128
8.6 The Form Effect...................................... 130
Symbols .........................................................132
References.......................................................135
Index.............................................................139
Chapter 1
Introduction
1.1 The General Problem
A ferromagnetic material may be defined as one that possesses a
spontaneous magnetization: that is, sufficiently small volumes of it
have a magnetization* (magnetic moment per unit volume) M8
dependent on the temperature but independent, or at least only
slightly dependent, on the presence or absence of an applied mag-
netic field.f The existence of this spontaneous magnetization is
explained by the Weiss “molecular field” postulate, amended
quantum-mechanically by Heisenberg; the amendment replaces
the mysterious molecular field by exchange forces, which are less
mysterious or more so according to one’s feeling toward quantum
mechanics. But this theory, based on exchange forces that tend to
align the spins and thermal agitation that tends to disalign them,
says nothing about the direction of the vector magnetization M;
only that its magnitude must be M8.
Experimentally, it is observed that though the magnitude of M
is uniform throughout a homogeneous specimen at uniform tem-
perature, the direction of M is in general not uniform, but varies
from one region to another, on a scale corresponding to visual ob-
servations with a microscope. Uniformity of direction is attained
* This word will be used instead of the longer term intensity of magnetization.
| By applied (magnetic) field we shall always mean the field of magnetizing
coils or magnets (or both) external to the specimen, as distinguished from the
field (be it the H field or the В field) produced by the magnetization of the
specimen under consideration.
only by applying a field, or by choosing as a specimen a body which
is itself of microscopic dimensions (a “fine particle”); the evidence
of uniformity in the latter case is indirect but convincing. The
tendency of a ferromagnetic specimen to break up into “domains/’
with their vector magnetizations oriented differently, explains the
possibility of a demagnetized state; and in fact such a domain struc-
ture was postulated by Weiss in order to reconcile his theoretically
predicted spontaneous magnetization with the experimental possi-
bility of demagnetization. Today the evidences of domain structure
are so many and so inescapable that its status is no longer that of
a postulate, but rather that of an experimental fact.
In two respects, however, the range of validity of this fact has
at times been supposed more universal than it actually is. *
First, domains were for a long time tacitly assumed to be present
in all specimens, regardless of their geometry. This naive assump-
tion delayed the theoretical understanding and practical applica-
tion of the properties of fine particles.
Second, domains have often been discussed as if they were a
phenomenon to be expected in all ferromagnetic malerials. Actually,
both theory and experiment indicate that domains in the usual
sense—regions within which the direction of the spontaneous mag-
netization is uniform or at least nearly so—do not occur unless
there are present strong “anisotropy” forces, which cause certain
special directions of magnetization to be preferred. When such
forces are absent or weak, the magnetization direction, over di-
mensions comparable with the usual domain dimensions, varies
gradually and smoothly.
It is therefore clear that domain structure, though normal, is
not universal. More generally, we should suppose merely that the
direction angles Ф and О of the spontaneous magnetization, or
equivalently its direction cosines a, 0, and 7 (subject to the con-
straint a2 + 02 + y2 = 1), are functions of the coordinates x, y, z
of the point P at which the vector magnetization M is being
evaluated. Whether these functions are constant or variable, con-
* See, for example, the discussions in Becker and Ddring (1939).
tinuous or discontinuous, step-functions or sinusoids, need not be
decided until later.
The general problem to be examined in this book is the problem
of developing a theory of this magnetic microstructure, concerning
which the Weiss-Heisenberg theory is noncommittal.
1«2 Forces Involved
The most basic method of solving this problem would be to use
an atomic model, such as a lattice of spins, and to introduce into
the model those forces that the Weiss-Heisenberg theory left out.
As has already been mentioned, that theory takes account only of
exchange forces and thermal agitation. The known forces that re-
main to be introduced are: magnetic dipole-dipole forces; forces
due to spin-orbit coupling and to magnetic quadrupole and higher
moments; modifications of the exchange forces that result when the
directions of neighboring spins are not exactly parallel; and “mag-
netostrictive” forces, which are not physically distinct from the
ones already enumerated but are the modifications of them that
come about because of the ability of the lattice to undergo strains.
Available methods of treating ferromagnetism atomically are al-
ready inadequate when only exchange forces and thermal agitation
are taken into account; they become quite unmanageable when,
for example, magnetic dipole-dipole forces are introduced. Accord-
ingly, we must resort to a phenomenological type of theory.
The possibility of such a theory rests on the fact that all these
new forces have only a small perturbing effect on the parallelism
(or, in certain cases, antiparallelism) of neighboring spins. The spin
direction, in other words, can change only by a small angle from
one lattice point to the next. It therefore seems legitimate to ap-
proximate the direction angles of the spins (or more conveniently
of. the associated magnetic moments) with continuous functions of
position. By this device, exactly analogous to the replacement of
individual atomic masses by a continuous density in elementary
mechanics, sums over lattice points are replaced by integrals over
a volume, and the techniques of calculus become applicable. The
basic concept in such a theory is a vector magnetization M whose
direction angles or direction cosines vary continuously with posi-
tion. Changes through appreciable angles may occur on a scale that
is small in comparison with the “domain0 scale, or on a scale that
is comparable with the domain scale; these two cases will cor-
respond, respectively, to the case in which “domains0 in the ordi-
nary sense are observed and to the case in which only a gradual
variation is observed.
The detailed development of such a theory is the theme of this
book. No claim is made that the theory has been fully developed;
all that can be said is that the foundations have been laid.
1.3 Methods of Solution
In the foregoing discussion, the term “force0 has been used in a
general sense. If magnetostriction is ignored, our model is a rigid
specimen with a vector magnetization M whose direction varies
continuously with the coordinates x, y, z, but whose magnitude
has a value M8 determined by the temperature. The “forces0 are
then torques (couples) that act on the magnetic moments Mdr of
the volume elements dr. In thermodynamic equilibrium, the
orientation of M at each point must be such that the total torque
on each moment element Mdr is zero. When the field is changed,
the torques in the old orientations usually cease to be zero; then
the dissipative processes that tend toward thermodynamic equi-
librium will ultimately establish a new equilibrium distribution of
orientations.
One method of describing the orientation of M is by use of polar
angles Ф, О referred to fixed x, y, z axes in the usual way. The cor-
responding components, L^dr and L^dr, of the torque on a volume
element dr are, in the dynamic sense, generalized forces (for the
element dr) corresponding to Ф and 9 as generalized coordinates.
In the course of our discussions we shall find it necessary to
recognize both the importance of temperature (since M9 and other
quantities are temperature-dependent) and the importance of
dynamic effects (since ferromagnetic resonance at microwave fre-
quencies is now a phenomenon of both theoretical and practical
significance). We therefore have need of a theory that is thermody-
namic in the literal sense of the word, i.e., capable of handling
thermal and dynamic effects simultaneously. Unfortunately ther-
modynamics in this sense does not yet exist; the body of theory
commonly known as “thermodynamics” is an equilibrium or at
most a near-equilibrium theory, more properly described as thermo-
statics. Our theory is therefore perforce a patchwork affair, proceed-
ing as follows.
In many of our discussions, the peculiarly dynamic effects
characteristic of resonance may be ignored, since the changes are
slow or the frequencies low. We may then pay proper attention to
thermostatic requirements. In such problems the convenient inde-
pendent thermal variable is absolute temperature T rather than
entropy S'. Therefore for a rigid material, the appropriate basic
energy function is not the internal energy U but the Helmholtz
function A; for a homogeneous specimen, at uniform temperature
and uniform magnetization, A = U — TS'. In a small reversible
change, for unit volume,
8U = ЛфЗФ + Le8B + T8S', (1-1)
- whereas
8A = ЬФ8Ф + Le8O - S'8T. (1-2)
Under nonuniform conditions and in the presence of an applied
field, more complicated expressions and additional transformations
are needed, and these will be discussed later; but from what has
been said, it is clear that the distinction between U and A is im-
portant, and that for isothermal processes (8T = 0) it is the latter
that plays the role of an energy function, from which the “forces”
(Аф, Ae) can be derived by differentiation with respect to the “co-
ordinates,” or from which the equilibrium values of the “coordi-
nates” can be found by a variational procedure.
In discussions of high-frequency phenomena such as resonance,
or of transient phenomena, dynamic contributions to the forces
are important. In such discussions we shall treat the specimen as a
dynamic system whose potential energy is of the same form as the
A of the equilibrium theory. If we actually identify this potential
energy with A, we thereby assume that the dynamic effects do not
appreciably perturb the thermodynamic equilibrium; in other
words, that the periods of the alternating fields, or the times re-
quired for establishment of magnetic equilibrium, are long in com-
parison with the times required for establishment of thermal equi-
librium. At the other extreme, we may identify the potential energy
with U if we assume an inequality in the opposite sense, so that the
conditions are approximately adiabatic rather than isothermal; the
parameters that appear in the energy formula must then, like the
“elastic constants” in vibration calculations, be interpreted as
adiabatic rather than isothermal “constants.” Use of our equations
with this interpretation will be legitimate only when the spontane-
ous magnetization M8 varies negligibly with temperature under
the conditions considered; for we shall always suppose that j M | =
M9 = const in the processes examined, and this is strictly true
only under isothermal and not under adiabatic conditions, since
M3 is a function of T and since in an adiabatic change T changes.
Unless one or the other of these simplifications—constant tem-
perature, or adiabatic conditions with negligible dependence of
M8 on T—is assumed, details of the heat-flow process must be
studied simultaneously with details of the magnetic process. We
shall not consider this complex situation. Over a limited range of
frequencies the departure from isothermal conditions, with result-
ing irreversibility, is taken into account by inclusion of a phe-
nomenological damping term in the equation of motion.
Though direct use of torques is sometimes convenient, energy
methods are generally more powerful. We shall therefore require,
as a starting point for most of our calculations, an expression for the
“free energy.” By “free energy,” as distinguished from “energy,”
we mean A or some other thermodynamic potential in which the
natural independent thermal variable is T, as distinguished from
U or some other thermodynamic potential in which the natural
thermal variable is S.
Besides torques and free energies, still another concept is some-
times useful; this is the “effective field.” If we choose as “coordi-
nates” not the angles Ф and G, but the components of magnetiza-
tion Mx, My, Mz, then the corresponding “forces” that are ob-
tained by differentiation of a free-energy density are quantities
3CZ, 3C,V, 3CZ with the physical dimensions of a magnetic field in-
tensity. We may regard the vector JC = 3Czi + JCyj + 3C*k, the
generalized vector force corresponding to M as vector coordinate,
as an “effective field intensity.” Since M is subject to the constraint
| M | = Me = const, the component of JC in the direction of M is
physically ineffective and in fact indeterminate. For by use of the
relation Mx2 + My + Mz = M?, a free-energy density of the
form /(Mx, My, Mz) can be replaced by, say, f(Mx, Mv, (Me2 —
Mx — My2)^); this yields on differentiation an К that differs
from the original one by a vector in the direction of M. From any
such JC, however, a unique vector torque per unit volume L,
normal to M, can be found by the formula
L = M X 3C,
(1-3)
just as in the case when JC is an actual external magnetic field in-
tensity. Similar remarks apply when the direction cosines a, (3, and
7 are used as coordinates: the corresponding force MS3C is indeter-
minate by an arbitrary vector in the direction (a, 0, 7), but the
torque is uniquely determined.
Whichever method—energies, torques, or effective fields—is
used in particular calculations, the first requirement is a method of
finding expressions for the various terms in the free energy, cor-
responding to the types of force already enumerated. Given such
expressions, the calculation of torques or effective fields is straight-
forward.
In Section 1.2 we classified the forces according to their physical
origin. In a phenomenological theory, it is more convenient to
classify them according to the mathematical form of the free-energy
expressions that describe them. In a rigid cubic crystal, the dipole-
dipole forces correspond to free-energy expressions similar in form
to the energy integrals of formal magnetostatic theory; spin-orbit
and quadrupole forces, to free-energy densities dependent on the
local direction of magnetization; and the exchange forces, as per-
turbed by nonuniformity of magnetization, to free-energy densities
dependent on the spatial gradients of the direction cosines (or direc-
tion angles) of the magnetization. These three contributions to the
free energy are usually called, in order, the magnetic or magneto-
static energy, the anisotropy or crystalline-anisotropy or magneto-
crystalline-anisotropy energy, and the exchange or exchange-stiffness
energy. The terminology is poor, for it confuses classification ac-
cording to origin and classification according to form; but in cubic
crystals it causes little trouble. In hexagonal crystals, on the other
hand,* the dipole-dipole forces contribute, besides the formal
magnetostatic-energy integral, a term in the form of a free-energy
density dependent on the local magnetization direction. In a phe-
nomenological theory, concerned with forms and not with origins,
this term must be treated as part of the “anisotropy energy” and
in fact cannot be distinguished from other terms of the same form
but of different origin; thus an energy term of magnetic origin is
included in the “anisotropy energy” and not in the “magnetic
energy.” We shall not try to reform the established terminology;
no confusion will occur if we remember that our theory is phe-
nomenological and that our classification is on the basis of form,
not of origin.
For a crystal capable of strain, the Helmholtz function contains
terms linear in the strains and terms quadratic in them. The latter
occur also with nonmagnetic materials and are called the “elastic
energy”; the approximation is usually made that the coefficients
in them are independent of the magnetization direction. (This does
not imply that the measured “elastic constants” are independent of
magnetization.) The linear terms have coefficients that depend on
the magnetization direction; these terms represent coupling be-
tween magnetic and mechanical processes and are usually called
“magnetoelastic” or “magnetostrictive” energy. After a trans-
formation to new thermodynamic variables, such as stresses rather
than strains, the new thermodynamic potential will again consist
of linear and quadratic terms; but the linear part of the new expres-
• * See equation (3-27) and the paragraph following it.
sion is not simply the linear part of the old expression rewritten
as a function of the new’ variables; therefore if the linear part is
again called “magnetostrictive” and the quadratic part “elastic,”
these terms now have new meanings. The conclusion from all this
is that descriptive names for individual terms in the free energy
should be used either with great caution or not at all.
1.4 Methods of Evaluating Energy Functions
If this theory were to be developed on the basis of a truly atomic
model, a conceivable procedure would be to evaluate the partition
function, and from it the free energy, by the methods of statistical
mechanics. Though formal expressions in the form of sums over
states and lattice sites might be derived without too much trouble,
the reduction of these to usable forms is out of the question. Fur-
thermore, even if this procedure could be carried out, it would not,
at least without basic modification, lead to a theory of magnetic
hysteresis; for standard statistical mechanics yields only states of
compete thermostatic equilibrium, and magnetic remanence is
not sdch a state.
The theory needed is a thermostatic theory based on phenomeno-
logical expressions for various contributions to the free energy, in
terms of such variables as may be relevant in the particular prob-
lems considered. These variables may include not only components
of magnetization and of elastic displacement, but also, as elasticity
theory illustrates, of their derivatives with respect to x, //, and z.
In the attempt to derive such free-energy expressions, there are
two possible approaches, which may be used singly or in conjunction.
One approach is to assume, at given temperature, a series in the
relevant variables, e.g. in the direction cosines a, d, 7; truncate
the series after a few terms, in the hope that these will prove suffi-
cient; and use crystalline-symmetry considerations to decrease the
number of (temperature-dependent) parameters in the formula.
This is the method usually used for evaluating the “anisotropy”
energy.
The other approach is to use an atomic model, perhaps dras-
tically simplified, to obtain an expression for a particular term in
the internal energy U at T = 0, where thermal agitation does not
complicate the calculation. The expression thus obtained may also
be considered an expression for A at T = 0. It may be adapted to
arbitrary T by replacing the constants in the formula by tempera-
ture-dependent parameters. This is the method that is convenient
for evaluating the contribution of dipole-dipole forces to the “mag-
netic” and “anisotropy” energies.
In either case, the temperature-dependent parameters in the
formula must be evaluated primarily by analysis of experimental
data; atomic models, however, facilitate the estimation of orders
of magnitude.
1.5 Scope of This Book
The primary aim of this book is to present, with reasonable
soundness and completeness, the basic principles and methods of
the theory just outlined. Illustrative applications will also be
described, but this aspect of the treatment will not be exhaustive—
partly because the details are too tedious for a book of this size,
and partly because not all of the attempts to apply the theory have
been equally fruitful. Attention will also be paid to the relations
between this theory, “micromagnetics,” and certain other branches
of ferromagnetic theory: specifically, domain theory, the theory of
ferromagnetic resonance, and spin-wave calculations. Here again
the treatment will make no attempt at exhaustiveness; rather, the
attempt will be to show to what extent a unified approach to these
various fields of study is possible and desirable.
Chapter 2 traces the historical origins of the theory. Chapter 3
summarizes the principles and methods to be drawn upon, and
Chapter 4 uses them to develop some of the basic equations of the
theory. The following three chapters apply these to specific classes
of problem: Chapters 5 and 6 to static problems (linear and non-
linear respectively), Chapter 7 to dynamic problems. In all these
calculations, the material is assumed to be rigid; Chapter 8 sum-
marizes and criticizes the usual methods of taking account of mag-
netostrictive phenomena.
Chapter 2
Historical Background
2.1 Early Concepts: Rotation and Inversion
The foundations of modern magnetization-curve theory were
laid by Akulov (1928, 1929a, b, 1930a, 1931a, b) and Becker (1930).
By laborious evaluation of lattice sums, Akulov derived the now
familiar formula for the crystalline-anisotropy energy density in a
cubic crystal,
F = K^a2$2 + fi2y2 + 7V), (2-1)
where'A i is a constant related to an assumed quadrupole moment.
In the range of positive field strengths H in which HMS is com-
parable with Kij the experimental magnetization curves of crystals
can be fitted quite well (with Ki evaluated from the data) by as-
suming that the magnetic moment of the whole specimen rotates
rigidly; its direction is then found by minimizing F — H*M at
constant H.
The rotation theory failed at small and negative H. When H is
along [100] in iron, the rigid-rotation model predicts a magnetiza-
tion M8 in the original direction until H reaches the negative
value — 2Ki/Ms. Experimentally, the curves are qualitatively
similar to those of polycrystals, with a coercive force two or more
orders of magnitude smaller than 2K\/M8. At low fields, therefore,
the “rotation” mechanism was assumed to be replaced by another
mechanism, called “inversion.” About the nature of this process
there was considerable doubt and argument. Akulov (1930b) for a
while maintained that in it the spontaneous magnetization ceased
ю exist. The view that ultimately prevailed, however, and that was
confirmed by visual observation of domain structures, was that the
ipontaneous magnetization persists locally but varies in direction
rom one region (domain) to another, so that the volume-average
nagnetization of the specimen can have a magnitude that is a small
raction of M9.
The rotation model was also applied, with only fair success, to
jolycrystalline material (Gans, 1932). The method of averaging
>ver orientations did not take proper account of magnetic inter-
ictions between the crystallites.
The rigid rotation model was used by Becker (1932), in conjunc-
tion with the concept of “internal stress,” to explain qualitatively
nany of the low-field properties of soft ferromagnetic materials.
The energy terms considered were those due to the applied field
md to the stress. The success of Becker’s theory was rather re-
narkable in view of its neglect of internal magnetostatic energies
arger than the energies it took into account.
2.2 Primitive Domain Theory
In Becker’s rotation theory, magnetic domain structure would
)e primarily a consequence of inhomogeneities of internal stress.
)ther authors, however, inclined to the view that domain structure
las a more fundamental origin. Exchange forces favor uniform
nagnetization without regard to its direction, whereas anisotropy
orces favor magnetization along special directions (e.g., the direc-
ions [100], [010], etc., in iron). The two classes of force together
avor uniform magnetization along some one of these special direc-
ions; they therefore cannot produce domains. If any of the classes
)f force discussed above is responsible for domain structure, it
nust be the dipole-dipole forces.
The interactions of an array of many dipoles present a com-
jlicated theoretical problem, and it is difficult to see whether their
general effect is to favor or to oppose parallelism of the dipole mo-
nents. If the two dipole moments in Figure 2.1 are constrained to a
lorizontal orientation, they will have less mutual potential energy
when parallel, as in Figure 2.1 (a), than when antiparallel; if they are
constrained to a vertical orientation, they will have less mutual
potential energy when antiparallel, as in Figure 2.1(6), than when
parallel; for in each case, the moment of each dipole then points
along the field due to the other. The energy is — 2m2/r3 in (a) and
—?n2/r3 in (6); therefore if the dipoles are free to assume arbitrary
orientations, they will prefer (a) to (b); but this does not say that
there may not be some third state that gives a lower energy than
Figure 2.1. Magnetic interaction of two dipoles.
either (a) or (6). Even this simple model presents a complicated
problem (finding the minima of a function of several independent
variables); and even this simple model demonstrates that whether
the dipole forces tend to favor parallelism or antiparallelism de-
pends on the constraining effect of other types of force.
Attempts to use megascopic* concepts are likewise inconclusive.
If we consider a prolate spheroid with longitudinal demagnetizing
factor ЛГ, and with a field Ho applied along its axis of symmetry,
and if we assume megascopic magnetization M in the direction of
Ho, we find that the internal megascopic field vectors are H =
Ho — УМ, В = Ho + (4тг — ЛГ)М. For Ho = 0, this becomes
H = — NM (field intensity of the poles to which the magnetization
is equivalent at external points), В = (4тг — N)M (field intensity
of the Amperian currents to which the magnetization is equivalent
at external points). Since 0 < ЛГ < 4ir, we conclude that the di-
* This term will be used, in preference to macroscopic, as an antonym to
microscopic.
pole fields oppose the magnetization or that they favor it, accord-
ing as we take H or В to represent the effective field of the dipoles.
Actually there is no reason for supposing that either of these mega-
scopic field vectors represents even an average effect of the dipole
moments associated with electron spins; but this argument does
illustrate the important fact that dipole effects are dependent on
the shape of the specimen.
On the basis of a lattice-sum calculation of Kornfeld, for a par-
ticular dipole array with zero total moment, Frenkel and Dorfman
(1930) assumed that magnetic dipole-dipole forces tend to produce
zero net moment in any volume element of the specimen. They then
attempted to calculate the domain size by estimating the increase
of exchange energy and the decrease of magnetic energy that result
when completely uniform magnetization is replaced by magnetiza-
tion uniform only within “droplets” of a certain size. They con-
cluded that the domain size depends on the specimen size and that
below a certain size, domains cannot exist. Both these conclusions
are accepted today. The details of the Frenkel-Dorfman calcula-
tion, however, are now of historical interest only.
In the 1930’s, most workers in ferromagnetism accepted the
existence of domains as a hypothesis necessary to reconcile mag-
netization curves with the existence of a spontaneous magnetiza-
tion. The hypothesis was supported by magnetic powder patterns,
the Barkhausen effect, and the Sixtus-Tonks (1931) experiments
on large regions of reversed magnetization; but it still needed to
be clarified theoretically. The domain-structure models used were
crude, arbitrary, and often not mutually consistent; they usually
neglected completely the complicated problem of magnetic inter-
actions between the domains.
In this state of vagueness, the domain concept was capable of
producing only a limited amount of quantitative theory. Becker
and Kersten (1930; Becker and Doring (1939), pp. 147 ff.) used the
rotation model to relate the initial permeability to other magnetic
and mechanical properties. Akulov (1931b) and Heisenberg (1931)
used primitive forms of domain theory to calculate curves of mag-
netostriction against magnetization; the results of these two cal-
culations were different but about equally consistent with the
experimental data. Of the two, Heisenberg’s lent itself more easily
to analytical formulation and generalization; it was therefore ap-
plied by other authors, with varying success, to a variety of
problems. This work has been summarized by Grimes (1957), who
recently revived the theory in an attempt to interpret reversible-
susceptibility data on ferrites. Akulov (1956) has also recently
revived his old theory. Neither of these theories has a clear and
convincing theoretical basis. The Heisenberg theory can be rein-
terpreted as an asymptotic form, valid when the number of pre-
ferred directions of magnetization is large, of a more general but
somewhat unrealistic theory based on the statistical theory of
extreme values (Brown, 1961a).
A quantitative domain theory, based on clearly formulated as-
sumptions and a definite model, became possible only with the
advent of a new concept: the displaceable interdomain wall.
2.3 The Wall Concept
The idea that magnetic processes depend in an essential way on
the properties of the transitional layer between adjacent domains,
and that it possesses a certain free energy per unit superficial area
in the same way that a liquid-liquid interface possesses free energy,
originated with Langmuir (Sixtus and Tonks, 1931). A theory of
the transitional layer and a formula for its thickness and its free
energy were worked out by Bloch (1932), and the interdomain
layer is commonly called the “Bloch wall.” Bloch, however, sup-
posed that the local spontaneous magnetization goes through zero
at the midplane of the wall. Wall calculations nowadays are based
on the model of Landau and Lifshitz (1935); in it, the magnetiza-
tion retains its magnitude M8 throughout the wall but changes
its direction with position along a path (say Oz) normal to the wall
surface. The magnetization direction is specified at z = — and
at z = +«>; the calculation yields formulas for the direction angles
at any z, for the wall “thickness,” and for the surface density of
free energy associated with the wall.
2.4 Early Calculations in Micromagnetics
The Landau-Lifshitz wall calculation was one of the first calcula-
tions in micromagnetics. It has been repeated many times, with
the modifications of detail required for special cases. We shall not
be interested in these, but rather in extensions of the micromag-
netics approach.
The general nature of a calculation in micromagnetics is as
follows. The total free energy of the system is expressed as a volume
integral. The integrand contains not only the direction angles of
the magnetization, but (in the exchange term) their spatial deriva-
tives. Therefore the search for a free-energy minimum leads to a
differential equation in the unknown angles or direction cosines;
it is in general nonlinear. In certain one-dimensional cases, such as
the wall calculation, the differential equation can be integrated
analytically. Usually, however, this is not possible.
One of the first calculations of this type was done by Bitter
(1936; 1937, p. 185). He took account only of anisotropy and ex-
ternal-field energies and assumed a variation with only one Car-
tesian coordinate. He obtained periodic structures described by
elliptic functions. Bitter noted that the maintenance of such a
distribution would require the presence of some special kind of
torque at the bounding surfaces of the specimen.
The Landau-Lifshitz calculation also leads to elliptic functions,
if boundary values of the angle are specified at finite distances
rather than at 2 = ±x. The more general periodic solutions were
studied by Shirobokov (1939, 1945) in calculations similar to
Bitter’s.
Three-dimensional problems were discussed qualitatively by
Elmore (1938). He summarized the formulas for the various terms
in the free energy, but he did not carry out the variational proce-
dure necessary for minimization. This was done in the general
three-dimensional problem by the present author (Brown, 1940b).
Solutions of the resulting equations were attempted only in the
linear approximation that is valid when the applied field is large.
Similar calculations were made by other authors (Holstein and
Primakoff, 1941; N£el, 1945,1948a, b), but with the exchange term
omitted—an approximation that considerably simplifies the dif-
ferential equations. The chief complicating factor in all such cal-
culations is the internal magnetostatic energy; it essentially inserts
an auxiliary potential problem, itself difficult, into a variational
problem already quite complicated.
The magnetostatic energy was taken into account in some of
these early calculations in micromagnetics. In domain theory,
meanwhile, it had been almost completely ignored.
2.5 Development of Modern Domain Theory
There are two parts to the domain problem: to explain the exist-
ence of domains and to calculate magnetic properties by use of
them.
As has been seen, domains must be attributed to the internal
magnetic (dipole-dipole) forces; yet that these indeed tend to pro-
duce domains is by no means obvious. If the formal expression for
the self-energy of the megascopic poles is accepted, rather naively,
as describing the actual interaction energy of the microscopic di-
poles, a demagnetizing tendency is at once evident; for the pole
energy can be written (81г)-1/H'2dr, where H' is the field intensity
of the poles, and this volume integral over all space achieves its
minimum value 0 when H' = 0 everywhere, i.e., when there are no
poles. But the replacement of atomic dipoles by continuous poles is
a step that requires justification. The justification of this step and
the conditions under which it is justified will be discussed in
Chapter 3.
If the pole-avoidance principle is accepted, the qualitative ex-
planation of domains runs as follows. Exchange forces tend to keep
neighboring spins aligned but have no direct long-range effect; the
extra energy involved in a nonuniform distribution of magnetiza-
tion directions depends only on the small angles between nearest-
neighbor spins, not on whether these small angles accumulate so as
to add up to a reversal within the specimen. Magnetostatic forces
are small in comparison with exchange forces for a pair of neighbor-
ing moments but are still important for moments separated by
great distances; consequently they cannot appreciably perturb the
short-range uniformity of the magnetization direction, but in a
large enough specimen they can select a nonuniform state that
avoids poles in preference to a uniform state that produces them.
The anisotropy forces now express their preference for certain
“directions of easy magnetization” by enlarging the regions mag-
netized nearly in such directions and contracting the regions mag-
netized in intermediate directions, so that the former become nearly
uniformly magnetized “domains” and the latter become thin
“walls.”
To make this argument quantitative and complete is one of the
central problems of micromagnetics, and it is one that has not yet
been solved satisfactorily. The stumbling-block is the nonlinearity
of the partial differential equations.
In most of the domain theory developed in the 1930’s, domains
were simply postulated; no attempt was made to explain their
existence. Becker’s rotation theory and the domain calculations of
Akulov and of Heisenberg were superseded by a theory in which
the mechanism of “inversion” was interpreted as a displacement of
the walls between domains; but quantitative calculations based on
this model were few and crude. The central concept in much of this
theory was still a vaguely defined “internal stress,” never reduced
to more fundamental concepts such as dislocations, although these
were then beginning to be used in the interpretation of plastic flow.
Internal magnetostatic energies were ignored. Despite these short-
comings, the theory achieved a certain degree of rounded develop-
ment and of success in fitting experimental data. At this stage it
was summarized in the book of Becker and Doring (1939).
The modern phase of domain theory, with its greater emphasis
on magnetostatic energy and on the details of domain geometry,
began with the now famous Landau-Lifshitz paper of 1935. Though
that paper was not cited in the Becker-Doring treatise, it had be-
come the basis of work by other authors (Elmore, 1938; Brown,
1940a, b). The paper contained two basic ideas that underlie all
present domain theory. The first is the idea of calculating the ef-
fective thickness and surface free-energy density of a wall, between
two domains of specified orientations, by a one-dimensional mini-
mization of the free energy. The second is the idea of selecting
domain structures on a larger scale by endeavoring to minimize the
sum of the total wall energy (the walls being now regarded as sur-
faces endowed with surface energy) and of volume-distributed
energies such as anisotropy energy. An important principle in both
these calculations is the minimization of the internal magnetostatic
energy by choice of magnetization distributions that avoid poles.
As an explanation of the origin of domains, the Landau-Lifshitz
reasoning is completely circular: in the calculation of wall proper-
ties, domains are postulated; in the calculation of domain proper-
ties, walls are postulated. But it does provide a plausibility argu-
ment. It furthermore is amenable to a consistency test, whose
far-reaching implications were overlooked for a decade; this topic
will be discussed in the next section.
It was a decade, too, before the Landau-Lifshitz ideas became
accepted by the main body of workers in ferromagnetism. This oc-
curred after 1945, when N6el’s further developments of the theory
became known, experimental colloid-pattern work strikingly con-
firmed some of his predictions, and theoretical work along several
new lines was stimulated. The theory at this stage was reduced by
Kittel (1949) to a body of doctrine that proved considerably more
permanent than the Вескег-Ddring doctrines of 1939. It is fair to
say that developments in domain theory since 1949 have related
mostly to details of specific problems and not to fundamental
notions. It is also fair to say that with one important exception,
the fundamental notions by which the theory of 1949 differed from
the Becker-Doring theory of 1939 had all been contained in the
Landau-Lifshitz paper of 1935. That one exception is the concept
of single-domain particles.
2.6 Single-Domain Particles
According to the 1930 calculation of Frenkel and Dorfman
(Section 2.2), sufficiently small particles should be uniformly
magnetized. This idea seems reasonable in view of the quantitative
explanation of domains outlined in Section 2.5: only the internal
magnetic forces favor nonuniform magnetization, and they become
effective only when they have megascopic distances over which to
operate. However, the Frenkel-Dorfman calculation was somewhat
obscure; and there was at the time no convincing experimental
evidence in favor of this, at the time, unorthodox idea.
The more precise and detailed Landau-Lifshitz calculation led
to a definite model for the nonuniform magnetic structure and to a
formula for the free energy of such a structure. It remained only to
compare this energy with that of uniform magnetization, and the
conditions for preference of the latter could have been determined;
but Landau and Lifshitz made no such comparison. When the com-
parison was made (Brown, 1940a), it led to the conclusion that a
sufficiently long cylinder should exhibit no domain structure; but
the implications of this conclusion were not realized. It was ap-
parently Kittel (1946) [but cf. discussion by Guillaud et al., 1953]
who first enunciated the principle that certain geometries should
result in uniform magnetization, with no domain structure. The
simplest example of the principle is the infinite cylinder (of any
diameter) with a direction of easy magnetization (minimum of
anisotropy energy) along the axis. Uniform magnetization along
the axis then minimizes all three terms in the free energy: the ex-
change energy, because the magnetization is uniform; the anisot-
ropy energy, because the magnetization is along a minimum of it
alone; and the internal magnetic energy, because there are no poles.
The state is therefore one of complete stability.
This argument, however, assumes absence of an applied field.
If a field is applied, opposite to the magnetization, eventually a
reversal of the direction of magnetization will occur. Whether the
magnetization will remain uniform during this (irreversible) re-
versal is a more complicated question, not considered in the stand-
ard “single-domain particle” theory of Kittel and others, and first
rigorously treated a decade later by the methods of micromagne-
tics; see Section 2.7.
The standard theory of single-domain particles is based on the
methods of Kittel (1946), Ndel (1947a, b), and Stoner and Wohl-
farth (1948). Like domain theory, it consists of two parts: determi-
nation of the conditions for existence of the single-domain state,
and calculation of magnetic properties on the assumption that the
particle is in that state. The first part of the theory is not rigorous;
the second is, if only isolated particles are considered. In practical
applications (fine-particle permanent magnets and magnetic re-
cording tapes), the magnetic interactions of many particles must
be considered; and here even the second part of the theory lacks
rigor. The theory has been reviewed recently by Wohlfarth (1959).
The standard method of determining conditions for a single-
domain state is to compare the free energy of the uniformly mag-
netized particle with the free energy of the same particle in some
assumed nonuniform state. The nonuniform state, as in the
Landau-Lifshitz domain theory, is conjured up out of the imagina-
tion of the theorist; it normally depends on some parameters, such
as domain dimensions, that can be adjusted for minimum free
energy. Thus the calculated free energy of the nonuniform state
depends on the ingenuity of the theorist. Furthermore the fact that
one of the two states considered has lower free energy does not
insure its occurrence: the higher-energy state may be metastable,
and then the height of the energy barrier between the states is
also important.* Finally, a particle that is single-domain in zero
applied field does not necessarily remain so in the process of mag-
netization reversal under the influence of a field; it is this latter
process that is significant in a calculation of the coercivity.
It was in the examination of this last problem that the methods
of modern micromagnetic theory emerged.
2.7 Recent Calculations in Micromagnetics
Let us consider again a particle initially uniformly magnetized.
To insure such uniformity, we begin with a large field Ho applied
* In fact, if in a changing field the particle were able to attain the lowest-
energy state at each instant, in defiance of these barriers, then there would be
no hysteresis.
along a direction of easy magnetization; let this also be the long
axis of the particle, which we assume to be an ellipsoid of revolu-
tion. If the field is now decreased gradually to negative values, the
state of uniform magnetization along the original direction is
always one of equilibrium; that is, the first variation of the free
energy vanishes for arbitrary small rotations, uniform or nonuni-
form, of the magnetization out of its original direction. The problem
is to find at what value of Ho the equilibrium becomes unstable:
that is, at what Hq the second variation of the free energy becomes
negative for some particular mode of rotation. If we can find this
HQ (the "nucleation field”), we can also find the mode of rotation
(i.e., the functional dependence of the small transverse components
of the magnetization upon the spatial coordinates) for which the
equilibrium is unstable. If this mode is a uniform rotation, the
particle perhaps remains single-domain throughout the cycle;
otherwise, it surely does not. The question of greatest practical
importance is whether or not the possibility of nonuniform rotation
modes makes possible a reversal at a smaller | Hq | than one would
calculate on the single-domain assumption.
A lower limit to | Hq | had been established (Brown, 1945) with-
out reference to the specific problem of single-domain particles;
but this calculation did not determine the mode of reversal. Kon-
dorskii (1952) investigated the problem by approximate methods;
he obtained formulas for the "critical diameter” of a sphere or
cylinder, defined as the diameter such that a particle of smaller
diameter reverses by uniform rotation, one of larger by nonuniform
rotation. A rigorous method of solving the problem was discovered
independently and almost simultaneously by the author (Brown,
1957b) and by Frei, Shtrikman, and Treves (1957). Subsequently
certain gaps in the original theory have been filled, and additional
problems have been investigated.
These rigorous calculations rest directly on the concepts and
methods of micromagnetics; there is no use of a "domain” or "wall”
concept as in the Landau-Lifshitz theory. There was also no use
of the domain and wall concepts in Kondorskii's earlier approxi-
mate calculation or in N6el’s (1947a) derivation of a single-domain
criterion: those also were calculations in micromagnetics, though
not rigorous ones.
The purpose of this book is to review the basic concepts and
principles that underlie these new, rigorous methods; and to review
also, though less exhaustively, the specific calculations that have
so far been made.
Ideally, one would hope by such methods to make present do-
main theory, with its incomplete arguments and often circular
reasoning, obsolete. Domains and walls, when they exist, should in
principle emerge from the theory without having to be postulated.
Practically, this is perhaps too much to hope and is at any rate
more than has actually been accomplished; for the linear problems
that can be solved by rigorous methods are special ones that do not
reflect the complexity of the usual physical situation, and the prac-
tically solvable nonlinear problems are even more restricted in
scope. Nevertheless micromagnetics has contributed some new
concepts that seem likely to prove valuable; and it can provide
precisely formulated methods of approximation and precise criteria
for assessing the approximations of domain theory. Quite possibly
the value of the micromagnetics approach lies as much in these
contributions as in its mathematical solutions of specific problems.
Chapter 3
Principles and Methods
3.1 Torque Equations, Static and Dynamic
Let Ldr be the torque exerted on the magnetic moment Mdr
of volume element dr (at r) by all the forces we have discussed.
This is the torque exerted on the magnetic moments (or on the spins)
of the electrons, not on the atomic lattice; its tendency is to rotate
the magnetization vector M with respect to the lattice. The condi-
tion for equilibrium of the magnetization direction is that at
everyr L = 0. (3-1)
The complexity of the problem lies in the fact that L is a compli-
cated function of the magnetization direction, the magnetizing
force H (or flux density B), and the first spatial derivatives of the
direction cosines of M; thus equation (3-1) is a partial differential
equation. The equation is furthermore nonlinear, and a number of
solutions may exist. Not all these equilibria are stable. The condi-
tion for stability is that an arbitrary small deviation from equilib-
rium must produce a torque that tends to restore the system to
equilibrium, not to increase the deviation still further. The boundary
conditions for the partial differential equation will be derived later.
The generalization of equation (3-1) to dynamic behavior is
dG/dt = L,
(3-2)
where Gdr is the angular momentum associated with the magnetic
moment Mdr. According to quantum theory,
G = М/уо, (3-3)
9.4.
where (in Gaussian units)
To = <?e/2mc;
(3-4)
e and m are the electronic charge and mass (e = — |e|), c is the
speed of light, and g is usually close to 2, since the moment is due
mostly to electron spin. Quantum theory is needed here only if we
desire a theoretical value for voj the phenomenological theory re-
quires only the concept that a magnetic moment entails a propor-
tional angular momentum. Elimination of G between (3-2) and
(3-3) gives the equation of motion of the magnetization,
dM/cft = ToL-
(3-5)
Solution of this equation involves all the complexities of the static
equilibrium problem plus the additional complexities of variation
in time.
From Section 1.3,
L = M X K, (3-6)
where JC is an “effective” magnetic field intensity. When the only
important term in К is a constant applied field intensity Ho, equa-
tion (3-5) becomes
dM/dt — ToM X Ho.
(3-7)
The right member is of the form co X M, with
co = -?0H0,
(3-8)
and (3-7) therefore describes a rotation of the vector M with
angular velocity co. This is the familiar Larmor precession of a mag-
netic top about the field direction. In general JC depends in a com-
plicated way on the magnetization in dr and at other points. The
“effective field” is therefore not usually useful for solving the dif-
ferential equations; but the concept of precession, suitably modi-
fied, is useful in physical interpretations of the behavior.
The equilibrium equation M X 3C = 0, which states that M is
along JC, can also be written
К - XM = 0, (3-9)
where X is an undetermined multiplier. This form occurs naturally
when the equilibrium condition is found from the free energy by a
variational procedure that uses Lagrangian multipliers.
3.2 Thermostatic Minimization Principles
Direct calculation of L or JC is usually more tedious than deriva-
tion of the equilibrium equation from the free energy. The free-
energy method has the additional advantage that it provides a
mathematically simpler stability criterion than that stated in Sec-
tion 3.1.
Figure 3.1. Calculation of magnetic work.
Consider a rigid specimen subject to an applied field, Ho, pro-
duced by current I in a filamentary circuit with line elements ds
(Fig. 3.1). To apply thermostatic theory, we must first derive an
expression for the work 617 done in an arbitrary small change.
The work is performed by the applied electromotive force in the
circuit and is —6/5/, where 8 is the induced electromotive force
— (l/c)d$/dZ; Ф is the part of the magnetic flux that is due to the
magnetization. Thus
(3-10)
where A is the vector potential due to the magnetization. Now*
A
M X r°
dr,
(3-11)
where г is the vector from dr to ds, r = | r|, and r° is the unit
vector r/r. Substitution of (3-11) in (3-10) gives
since the expression in curly brackets is the applied field Ho at the
position of dr. (Here M = dM/d/ is the time rate of change of M
at a fixed point.) Therefore
5Ж
5Mdr,
(3-13)
and the first law of thermodynamics for this system becomes
bU
(Hq*6M + 3Q)dr,
(3-14)
where bQdr is the heat absorbed by volume element dr from ad-
jacent elements.
When M is constrained to a constant magnitude M9) 5M must
be of the form 50 X M, which represents a small rotation of amount
* See, for example, Panofsky and Phillips (1955), p. 127.
| Зв | about the direction of the vector Зв. Then Ho* 3M = Ho* Зв X
M = M X Ho* Зв; this represents the work done in the small rota-
tion Зв by the torque M X Ho.
We assume that a physically small volume element may be con-
sidered to be at a definite Kelvin temperature T, though the tem-
perature need not be uniform throughout the specimen, and that
the entropy S'dr of a volume element is definable in all states con-
sidered. Then the second law of thermodynamics states that*
3Q < T8S'f
(3-15)
with the inequality sign in natural (irreversible) changes and the
equality sign in reversible changes. Substitution of (3-15) in (3-14)
gives
Let
and
8U
3M + T8Sf)dr.
(3-16)
(3-17)
Ho-Mdr =
; (3-18)
then A is the Helmholtz function and G the Gibbs function (both
‘free” energies). In terms of G, the second law becomes
3G
M-5H0 - S'8T)dr.
(3-19)
Now let the temperature T and the applied field Ho have def-
nite values, with T uniform so that a state of thermostatic equilib-
rium is possible, but with the specimen not necessarily in such a
state. Then (3-19) reduces to
3G < 0
(3-20)
* If q is the heat current density, 8Q — — (V*q)5t The inequality (3-15) is
therefore equivalent to TS' > —V*q. This inequality can be obtained from
he more usual one JS'dr > —J(n’q/T)dS, or JS'dr = —J(n*q/T)dS +
(dS'/dt)i„dT, by inserting the value q*V(l/T) of the part of (dS'/di) that
в due to heat flow. Cf. Tolman and Fine (1948), equations (18.4) and (8.4).
and states that in natural changes, G can only decrease. The condi-
tion for stable equilibrium is therefore that G must be a minimum
with respect to changes consistent with the constraints дТ = 0,
5H0 = 0.
Thus if we have a formula for G as a function of Ho, T, and in-
ternal coordinates (such as the direction cosines of the magnetiza-
tion at each point), we can find the stable equilibrium values of the
internal coordinates by minimizing G with respect to these coordi-
nates. Usually the number of minima exceeds one, and therefore
the state of magnetization is not uniquely determined. If | Ho | is
sufficiently increased, the aligning effect of Ho dominates and re-
duces the number of stable equilibrium states to one; on subsequent
decrease of Ho to its original value, the system may find itself in a
state different from the initial state. This principle accounts for
hysteresis; it is illustrated most simply by the theory of uniaxial
single-domain particles (Stoner and Wohlfarth, 1948). For speci-
mens capable of a domain structure, we must expect a multitude of
stable equilibrium states. Over sufficiently long time intervals,
transitions between them occur; an equilibrium statistical-mechani-
cal calculation, if it could be carried out, would give the long-time
average over the various states. But the time constants for transi-
tion are so long that for most purposes the equilibrium of any one
stable equilibrium state may be regarded as permanent.
Our procedure for finding a minimum of G will be as follows. We
seek first an expression for A, which may be regarded as the internal
“potential energy/’ in the mechanical sense, under isothermal
conditions. The internal coordinates that appear in this expression
will be the direction cosines a, /3, у or (i = 1,2,3) of the magnet-
ization (or equivalent angles), at all points г in the specimen, and
their spatial gradients Vat-. Our methods of arriving at such expres-
sions will be partly microscopic and partly phenomenological, as
was explained in Section 1.4; but the expressions, once obtained,
will be regarded entirely phenomenologically. The assumption at
this point is that the specimen is in a state of stable internal equi-
librium for the given magnetization distribution: that is, if the
specified values of щ were maintained by appropriate applied fields
or other external forces, no changes such as heat flow, lattice dis-
tortion, or diffusion of atoms would occur. The specimen is not,
however, necessarily in external equilibrium with the actual applied
field. To find the values of a, compatible with such equilibrium, we
add to A the term — JHO-Mdr, in accordance with (3-18), to find
G as a function of the internal variables a» and Vat-, and then mini-
mize the resulting expression with respect to these variables.
The applied field appears only in the added term - jHo-Mdr.
When Ho is uniform, the added term reduces to — Hoem, where m
is the total magnetic moment of the specimen; in any case it repre-
sents the potential energy of the magnet of moment m in the fixed
field Hq. It is the analog of the term pV that is added to the internal
energy of a fluid to get its enthalpy or to its Helmholtz function to
get its Gibbs function. As A may be considered the “potential en-
ergy/’ under isothermal conditions, of a system consisting of the
specimen alone, so G may be considered the “potential energy,”
under isothermal conditions, of a system consisting of the specimen
plus an ideal permanent magnet that produces the field.
For a deformable specimen, other terms must be added to the
work done, and the free energy to be minimized may be a different
one. In general, the thermodynamic potential to be minimized, in
the condition for stable equilibrium at specified values of variables
is the thermodynamic potential whose small reversible change
contains the differentials tyi.
The process of finding a minimum of G consists of two steps: find-
ing a state for which the first variation of G vanishes, and showing
that for this state the second variation is positive, in each case for
arbitrary variations of a, 0, у consistent with a2 + /32 + y2 = 1.
The first step selects the equilibrium states, and the second tests
their stability.
3.3 Dynamic Variational Principles
In dynamic problems we must replace equation (3-1) by equation
(3-5). If the dynamic process is reversible and isothermal, the
torque will be given by the same expression as in equilibrium, and
all that is necessary is to replace 0 by y0 ММ/Л on the right of
(3-1).
This method of transforming the equilibrium equation to an
equation of motion usually suffices. For some purposes, however,
it is convenient to derive the equation of motion, like the equation
of equilibrium, by a variational principle.
For a reversible specimen, the appropriate principle is Hamilton’s
principle* .
5 I £dt = 0, (3-21)
Л1
where £ is the Lagrangian function, and where the generalized
coordinates are held fixed at times ti and To derive an expression
for <£, consider first a small magnetic particle of volume v, magnetic
moment Му, and associated angular momentum Gv = My/y0,
and with potential energy У(Ф, G), where Ф and О are the polar
angles of the direction of M. We may regard this as a classical top
with principal moments of inertia A, A, C and coordinates Ф, G, Ф
(Euler’s angles), in the (classically impossible) limit A = 0, C > 0.
The third Euler angle, Ф, is the angle of rotation of the top about
its axis of symmetry from a certain standard configuration.! The
Lagrangian function of this system is
£0 = |С(Ф + Ф cos G)2 - Г(Ф, G). (3-22)
The Lagrangian contains Ф but not Ф; Ф therefore is, in Routh’s
terminology, an “ignorable” coordinate and can be eliminated by
the procedure known as “ignoration of coordinates.”! This pro-
cedure yields the reduced Lagrangian or “Routhian”
• d£n •
<£ = £o — Ф —— = const + cos G — У(Ф, G), (3-23)
0Ф
* See, for example, Whittaker (1927), pp. 245-247, or Goldstein (1950), pp.
30-38.
f See, for example, Whittaker (1927), pp. 9-10,155-156, or Goldstein (1950),
pp. 107-109, 164-165. The present Ф, 0, Ф are Whittaker’s Ф, 0, which are
related to Goldstein’s ф, 0, ф in the way explained by Goldstein.
t See Whittaker (1927), pp. 54-57; Goldstein (1950), pp. 47-49, 218-220.
where До is the constant value of the angular momentum д£0/дФ
corresponding to the coordinate Ф. The Lagrangian equations of
motion derived from (3-23) are equivalent to (3-5) if До is assigned
the value vMB/yQ, the angular momentum associated with the mag-
netic moment of the particle.
For an extended specimen, we replace До = »MB/yQ by (MB/yo)dr
and integrate, then replace / Vdr by the total free energy G. Thus
MB r.
£ = — | Ф cos Gdr - G s T - G. (3-24)
y^J
The first term in (3-24) appears to assign a special role to the direc-
tion of the polar axis Oz, from which 0 is measured; but all trace
of such a special role disappears in the contribution of this term to
the equation of motion. Thus the special direction Oz may be chosen
arbitrarily. The form of £ is therefore not unique. The difference
between two possible expressions is the time derivative of a func-
tion of the coordinates and contributes nothing to the equation of
motion; for in (3-21) its variation integrates to zero.
Since the kinetic term in (3-23) is entirely gyroscopic, i.e., linear
in the generalized velocities, it contributes nothing to the energy
function, which accordingly is simply the potential energy V.
Similarly, for the extended system, the energy function is simply G.
At constant Ho, it is this function that is conserved when there is no
dissipation and that decreases with time under the influence of
dissipative forces. Thus the joint precession of all the spins occurs
along a locus of constant G or decays toward one of lower G, ac-
cording as there is not or is dissipation.
3.4 Free-Energy Formulas
We now require explicit expressions for the various terms in the
Helmholtz function A.
To evaluate the magnetostatic term, we use the microscopic
method. For a lattice of dipoles at T = 0 the potential energy is
where h't- is the field intensity at the position of dipole i due to all
the other dipoles. We assume that from one dipole to the next the
direction of mt- varies so slowly that over a physically small
sphere,* the components тгх, тг1/, may be taken to be either
constants or at worst linear functions of the position coordinates of
the dipole г; because of the strong exchange forces that couple one
spin to the next, this should be a good approximation in ferromag-
netic materials. Then by the well-known argument of Lorentz
(1909, pp. 137-139 and Notes 54-55), f
h't = H + H + h'\, (3-26)
where H' is the megascopic field intensity computed from the poles
due to M, and where h", is the field of the dipoles within a physi-
cally small sphere about dipole г. For a cubic lattice, h"t- = 0; for a
crystal lattice in general, h", = A*M, where Л is a tensor whose
* A “physically small” sphere is usually defined as one whose radius is large
in comparison with the lattice spacing but still small on the scale of ordinary
observations. Here we must replace the second requirement by the more strin-
gent one that the radius be small on the scale of domain observations. In fact,
we should like it to be small in comparison with the thickness of a domain wall;
and when we cannot satisfy this last condition, we should have some misgivings
about the use of the Lorentz formula at points inside a wall. We can, however,
usually satisfy all these conditions reasonably well by taking the radius to be
about 10 lattice spacings.
t The Lorentz formula (3-26), with h“* = A*M, is obtained by starting
with the magnetizing force H' or flux density B' computed from a continuous
magnetization, subtracting from it the contribution H" or B“ of the continuous
magnetization within the Lorentz sphere, and adding instead the field of the
actual dipoles located at lattice points inside the sphere. If the magnetization
is uniform inside the sphere, H" = — $irM and B" =» 4- $irM; thus the sub-
traction gives H' + -jirM or its equal, В' — -yirM. If the dipole moments within
the sphere are all equal and parallel (an assumption consistent with uniformity
of M), their field is of the form A*M, where Л has trace zero and in a cubic
crystal vanishes. If we now permit linear variation of the components of M
with the coordinates and, consistently, linear variation of the components of the
moments with the lattice-point coordinates within the Lorentz sphere, the
resulting additional terms in h\ integrate or sum to zero; but they no longer do
so if a quadratic variation with coordinates is allowed. This is the reason for
the restriction to variations no worse than linear over a physically small sphere.
form is determined by the crystal symmetry and whose trace is
zero. On substituting (3-26) in (3-25) and replacing sums by in-
tegrals, we get
Um = - J Jm • (H' + f тгМ + Л • M)dr. (3-27)
Here M is the magnetization at the temperature considered, namely
(so far) absolute zero, for which A = U.
We now assume that at any temperature T, A contains a term
of the form (3-27), with M the magnetization at temperature T.
Then M*M = M2, a constant at given T; therefore the second
term in (3-27) is constant in our variational procedures and may
be dropped. Finally, the integrand M*A*M is of the form of a free-
energy density dependent only on the local direction of the mag-
netization; it may therefore be absorbed into the “anisotropy” term
in A, Thus
= -jjMH'dr. (3-28)
The use in (3-28) of the pole field H', rather than the Amperian-
current field В' = H' + 4тгМ, is wholly arbitrary. Substitution of
H' = В' — 4тгМ in (3-28) gives a constant plus
A'n = -jJ'w-B'dr. (3-29)
The choice between these two formulas is one of convenience alone.
To evaluate the anisotropy term in A, we use phenomenological
methods. We assume a power series in a, /3, and y, use the crystal
symmetry to decrease the number of coefficients, and truncate the
series after the first two nonconstant terms. This gives for cubic
crystals an anisotropy energy density
wa = Ki(a202 + fi2y2 + y2a2) + K20t2fi2y2
(3-30)
where the cubic axes are chosen as coordinate axes; and for hex-
agonal crystals
wa = Kx(l - V2) + #2(1 - V2)2,
(3-31)
where the hexagonal axis is chosen as the z axis. The constants Ki
are functions of T.
To evaluate the exchange term in A, we use again a microscopic
model. We consider only nearest-neighbor interactions. We write
for the exchange energy of two nearest-neighbor spins
W* =
-2JSySy,
(3-32)
where J is a constant and Sfi is the spin angular momentum of
spin i. We assume that the angle 0ty between S, and Sy is very small;
then Si-Sj = S2 cos 0t7 == S2[l - |0O2] = S2{1 - ||vy - vt-|2},
where v< is the unit vector along — St- (and therefore along the as-
sociated magnetic moment), and where S = ISJ = |Sy|. We as-
sume further that vt may be approximated sufficiently with a con-
tinuous function v of position; then v can be interpreted mega-
scopically as the unit vector along M, i.e.,
v = ai + /3j + 7k. (3-33)
If sy is the position vector of spin j with respect to spin г, vy — v/ =
sy Vv in the approximation assumed, and the excess energy due to
the nonparallelism of Sz and Sy is
W/y = JS21 Vy - Vi I2 = JS2(sy• Vv)2. (3-34)
If there are n spins per unit volume, the density of the excess ex-
change energy is
we = %nJS2 E (s>-Vv)2, (.3-35)
3
where the sum is over nearest neighbors.
For a cubic crystal, this gives
we = |C[(V«)2 + (V/3)2 + (V7)2] (3-36)
with
1 , , 2JS2c
C = - nJS2 У. S;2 =--------- (3-37)
3 у a
where a is the length of the edge of the unit cell, and where c = 1, 2,
and 4 for simple, body-centered, and face-centered cubic lattices,
respectively.
For a hexagonal crystal,
with
Ci = yjs2 E p/>
J
C2 = nJS2 E z/;
i
(3-38)
(3-39)
(3-40)
Pi is the projection of sj in the basal plane, and zy is its projection
along the hexagonal axis. For ideal close packing, as in cobalt, this
reduces again to (3-36), with
C = ±nJS2a2 = 4V2 JS2/a;
(3-41)
a is the distance between nearest neighbors. Thus formula (3-36)
covers most of the cases of interest.
As before, we now interpret (3-36) as a formula for the contribu-
tion of the nonuniformity of magnetization to the exchange term
in the free-energy density at any temperature, with C possibly
temperature-dependent.
In both the microscopic arguments used above, the dipole or spin
considered is assumed to be far enough from the surface so that in
the magnetic calculation all the other dipoles within the Lorentz
sphere, and in the exchange calculation all the nearest neighbors,
are actually present. The phenomenological anisotropy argument
likewise invokes a symmetry dependent on completeness of the
environment on an atomic scale. Lattice sites near the surface re-
quire a special treatment.
In the Lorentz field calculation, the error consists in the incorrect
inclusion of the fields of lattice dipoles and megascopic volume and
surface charges located in the part of the Lorentz sphere outside
the specimen. These are short-range effects. Therefore the “mag-
netic” as well as the “anisotropy” surface contribution can be de-
scribed by a surface density of free energy, and the sum of the two
can be treated phenomenologically. In a first approximation, we
may neglect the dependence on the orientation of the surface
normal n = Zi + mj + nk with respect to crystal axes and suppose
that the surface energy-density depends only on the orientation of
M with respect to n. Since reversal of all the dipole moments does
not change any of the microscopic energy terms we have considered,
the leading term in the surface anisotropy energy density is
w8 = |#e(n*v)2 = %K8(la + m/3 + ny)2. (3-42)
For a cubic crystal, the dependence of w8 on n can be taken into
account by replacing (3-42) by
w8 = |Kel(Z2»2 + m2fi2 + n2y2) + K82(bnafi + winffy + nlya),
(3-43)
which reduces to (3-42) when K82 = K8i.
N6el (1954) has estimated the surface anisotropy energy (3-42).
By dimensional reasoning | K81 = VtvMe2, where I' has the dimen-
sions of length; and according to N6eFs estimate, V = 102A. There-
fore the surface anisotropy should ordinarily affect conditions only
within a distance of this order from the surface. We shall include
the К8 term in our general equations but shall neglect it in most of
the specific applications.
In the exchange energy calculation given above, the only spins
affected by surface asymmetry are those right at the surface. In-
stead of summing over all spins and, for each spin, over its nearest
neighbors, and then dividing by 2, we can proceed as follows. From
each spin site, a line can be drawn to each nearest-neighbor spin
site; each such line can be extended through the crystal and will
then contain a large number of uniformly spaced spin sites. Form
all such lines. For each line, sum the energies of all pairs of suc-
cessive spins; for each group of parallel lines, sum the contributions
of the different lines; approximate the sums so far formed by in-
tegrals; finally sum over the various orientations of the lines. By
this method we conclude that to within the errors involved in the
replacement of sums by integrals, the total energy is given cor-
rectly by integration of the expression (3-36) or (3-38), without a
surface term, unless the lattice geometry and surface orientation
are such that differently oriented nearest-neighbor lines systemati-
cally terminate in different planes.* We ignore such special cases.
Absence of a surface term in the energy, however, does not imply
that there can be no surface exchange torque. Such a torque clearly
can exist; for when the spins are not quite parallel, a surface spin
is subject to a torque from its inner neighbor and not from its outer.
The surface torque equation, however, will emerge from the volume
exchange-energy integral in the variational procedure, just as the
surface force equation emerges from the volume elastic energy in
the variational procedure of elasticity (Love, 1934, pp. 166-167).
On collecting all terms in A and adding the term that transforms
from A to G, we get
G = f {|C[(V«)2 + (W?)2 + (VT)2] +wa~ - M-H0]dr
where wa has the form (3-30) or (3-31), as may be appropriate.
3.5 Magnetostatic Theorems f
In dealing with the magnetostatic self-energy, certain transfor-
mations are useful. These can be derived from the general theorem
that if H' is irrotational everywhere and B' is solenoidal everywhere,
and if the corresponding scalar and vector potentials Ф' and A' are
“regular at infinity,” the volume integral through
all space vanishes. “Regularity at infinity” means that |ф'| and
| A' | must vanish at infinity at least as fast as 1/r, and | Гф' | and
* A two-dimensional example of this situation is a lattice of equilateral tri-
angles with the spins at the vertices, and with a boundary bisecting a triangle
edge: one set of nearest-neighbor lines perpendicular to the boundary sys-
tematically terminates a half-edge farther in than the other such set and than
the lines in other orientations.
f For a more detailed discussion of these theorems, see Brown (1962c),
Chapter 3.
| V X A'| at least as fast as 1/r2; these conditions are satisfied
when H' is the magnetizing force and B' = H' + 4тгМ the flux
density due to the magnetization M of any finite body. On apply-
ing the theorem to this case, we get
(3-45)
The left member is Am of (3-28). Since the right member is positive
unless H' = 0 everywhere, and since this is possible only when all
the volume and surface pole densities — V-M and n-M vanish,
we see at once that the magnetic self-energy is minimized when
there are no poles. Thus we have justified—to within the accuracy
of the Lorentz local field approximation—the pole-avoidance
principle mentioned in Section 2.5. The alternative expression
A'm can be transformed to — (1/8тг)/В'2</т; because of the minus
sign, this form leads to no simple principle, in terms of Amperian
currents, analogous to the pole-avoidance principle. For this reason
H is a more useful field vector than В in micromagnetics.
More generally, let Mi and M2 be two distinct magnetization
distributions, occupying regions that may or may not coincide;
let H\ and B\« = + 4тгМг- be the magnetizing force and flux
density produced by Мг. Then since the integrals of H'^B'2 and
of H'2*B'i vanish, we get
- fH'pMjdr = 7~f h'i h'2<Jt = -fН'г-Мк/т.
(3-46)
This reciprocity theorem has the following two especially useful
applications:
(a) Let <5M be a variation, actual or virtual, from initial magneti-
zation M, and let 5H' be the resulting variation in the magnetizing
force due to M. Then
H4Mdr =
M-SH'dr,
(3-47)
and therefore the first variation of Am is
SA
+ M3H
')dr
н'-амйт.
(3-48)
(b) Formula (3-48) is valid only to the first order in 5M. For
some purposes we need an expression valid to higher orders in the
change (actual or virtual) from some reference state. Let M(0) be
the magnetization in the reference state, AM the change of mag-
netization; let H'(0) be the magnetizing force due to the magnetiza-
tion M(0) in the reference state, AH' the change of magnetizing
force due to the change AM.* Then the change of Am from its
reference value is
'(0)-AM + AH'-M(0) + AH'-AMldr
Гн/(0) • ДМЙТ + L J (4H')2dr.
(3-49)
This is exact. In many applications, however, A4m is needed only
to the second order in the independent variables that describe the
change AM. In particular, suppose that M(0) = M8k is a uniform
magnetization along Oz, that AM = 2l/e[ai + Pj — (1 — y)k] is
a small deviation from this uniform magnetization, and that AAm
is needed only to the second order in a and /3. Then AM in the first
integral will be expressed to the second order, namely as Me[ai +
P5 ~ i(«2 + 02)k], where the term Me[ai 4- 0j] is of first order
and the term — |Л/8(а2 + 02)k is of second; and AH' in the second
* In the present and later discussions, the superscript indicating “refer-
ence state,” must not be confused with the subscript o, which distinguishes the
applied field Ho from the total magnetizing force H and from the part H' of it
that is due to M.
integral will be expressed to the first order, namely as the magnetiz-
ing force H'(1) due to magnetization + 0j].*
The reader is assumed to be familiar with the formulas for uni-
formly magnetized ellipsoids and with the concept of demagnetiz-
ing factors, which can be extended to any uniformly magnetized
body (Brown and Morrish, 1957; Brown, 1960).
3.6 Variational Techniques
To find equilibrium conditions, we imagine the magnetization
to undergo virtual variations 5M, subject to the constraint M2 =
Ms2 but otherwise arbitrary, and require that the resulting first
variation of G be zero. Explicitly, we set a = a0 + euy +
er, 7 = (1 — a2 — 02)^ and require that and have such
values that dG/de vanishes at a — a0 and 0 = 0o for arbitrary func-
tions u(r) and v(r). This precise formulation of the principle can be
dispensed with when we are interested only in equilibrium and not
in stability conditions; it is simpler just to expand the variation 8G
to the first order in 5M and then to require that SG vanish for ar-
bitrary 5M consistent with the constraint M • 5M = 0. There are
two methods of imposing this constraint:
(a) We can set 5M = 50 X M, where 50 is an arbitrary small
vector; this describes a rotation of M through a small angle 1501
about an axis in the direction of 50. We may then set the coefficients
of the components of 50 perpendicular to M equal to zero (the
parallel component is obviously ineffective). This method gives di-
rectly the torque equilibrium equation (3-1) (or its surface counter-
part) , a vector equation equivalent to two scalar equations (since
L = M X 3C is perpendicular to M).
(6) We can require that 8G +/X(r)M*5Mdr +/д(г)М-5М
dS vanish for arbitrary 5M; here the Lagrangian multipliers X and
д are unknown functions of position in r and on S. This gives the
force (or effective field) equilibrium equation (3-9) (or its surface
* The superscript will be consistently used to label a quantity that is of
order n in the small quantities a and (3.
counterpart), a vector equation equivalent to three scalar equations
but containing the undetermined function X (or д). Elimination of
X (or m) gives the torque equilibrium equation.
In the evaluation of the variation 8G, the volume and surface
anisotropy terms require no special tricks. The variation of the
magnetic term can be expressed in a form involving only 5M (and
not 6H') by the method described in the preceding section. The
variation of the exchange energy involves such quantities as 8 Va =
to get rid of we do a three-dimensional integration by
parts:
• V 8adr
{V • [(Va)3a] — (V2a)3a|dr
da
— 8adS - С I (y2a)8adr. (3-50)
dn J
Thus we finally express 8G as a volume integral plus a surface in-
tegral, each containing no variations except 3M = Ms8v =
7lfe(i5a + j8P + k&y). Either of the two techniques described
above can then be used to obtain the volume and surface equi-
librium equations.
In the application of Hamilton’s principle, T of equation (3-24)
may be written
T =J3dr, (3-51)
where
Ms . M, у
3 = — Ф cos e =---------------- (aj9 - /За). (3-52)
T(. Vo Vo 1 — V
If
д д д д д д д d
— = i-------H j-----h к —• > — = i-------f- j —г + к — (3-53)
dv да <3/3 dy dv да <3/3 ду’
then
(3-54)
or by integration by parts with respect to time
• 5vdr. (3-55)
After transforming the — G term as in the equilibrium calculation,
we may apply either of the techniques used before to obtain an
equation valid at each instant and at each point of r or 8. This
gives the dynamic generalization of (3-9),
1 / d аз d3\
—---------г + — ) + ГС = ХМ,
Мe \ dt dv dv/
(3-56)
where, if a, a, 0, and 0 are taken as the independent variables in
(3-52),
d аз аз м9 - pi
dt dv dv To 7
Ms V X к
7o 7
(3-57)
Application of MX to (3-56) then gives equation (3-5), since
vX (v X k) = v* к v - v* v к = yv.
3.7 Treatment of Dissipation
The dynamical processes have so far been supposed reversible.
In a phenomenological theory, the simplest procedure for adding
a dissipative term is the following. If the Cartesian components of
M are regarded as generalized coordinates (for unit volume), then
the corresponding generalized external forces are the components
of Ho, since the work done in a small change is Ho • 5M. Therefore
the generalized forces with internal contributions included are the
components of ГС. To these we now add dissipative terms — y]MX)
—qjjy, —proportional to the generalized velocities, with ?? a
positive constant. (Unless the dissipative process is known to be
anisotropic, we have no reason to use three different constants
*12, Пз-) This adds to the effective field JC a term — ijM and to the
torque a term — ijM X M. The equation of motion now becomes
dM
----= 70M X (3C - i?M)
dt
= 7o(L - X M), (3-58)
where JC and L are the effective field and torque derived from the
free energy. Equation (3-58) is the equation of motion proposed by
Gilbert (1955). For given 70 and 7? it is equivalent to the older
Landau-Lifshitz (1935) form
dM X
= 7'0M X 3C -M X (M X K); (3-59)
dt-------------------------------------------M2
but if 7o is assumed to remain constant as ?? varies, then 7'0 will
not remain constant. We shall use the form (3-58). In the study of
thermal fluctuations one may add to JC also a fluctuating term
JCrandom? whose (statistical or time) average is zero, and whose
statistical properties are similar to those of the fluctuating force
in the theory of Brownian motion; then (3-58) becomes the Lange-
vin equation of the stochastic process (Brown, 1959b).
The dissipative torque can be derived from a Rayleigh dissipa-
tion function
F =f^r = J J\M2dr. (3-60)
This contributes a term to the generalized force correspond-
ing to Mx, i.e., to 3CX, and hence a term —tjM to JC and a term
—X M to L.
With dissipation of this type, the variational principle from
which the equation of motion can be derived is
ph ph p $3
6J £dt - J dtJ —-bvdr = 0. (3-61)
The resulting equation of motion is
d dZ dZ
--------1----1- MS3C-----
dt dv dv dv.
(3-62)
where 3C is the same as in equilibrium.
The Lagrangian formulation of this problem and the use of a
Rayleigh dissipation function are results of the work of Gilbert
(1955).
Chapter 4
Basic Static and Dynamic Equations
4.1 Equilibrium Torque Equation
The equilibrium condition may be written either in the torque
form (3-1), with the torque L of the form (3-6), or in the “effective
field” (force) form (3-9). This condition must hold at each point
in the volume r of the specimen, and an analogous condition in-
volving the torque or force on unit area must hold at each point on
the surface S. Formulas for L or JC and for its surface analog are
most easily derived by variation of the appropriate free energy. We
shall now carry out this variational procedure for a rigid specimen,
for which the appropriate free energy is G, equation (3-44). The
variational techniques necessary have been outlined in Section 3.6.
For derivation merely of the equilibrium condition, as distin-
guished from conditions for stable equilibrium, we retain only terms
of first degree in the small variation 3M = M8dv = M88 (ai +
Pj + 7k). This is to be an arbitrary small variation consistent with
the constraint M2 = M82 (or v2 = a2 + /32 + y2 = 1) and with
continuity and differentiability of a, ft, and 7 and of their spatial
derivatives. The resulting variation of G is
ZG
{C[(Va).5(Va) + (V/3)-5(V/3) + (V7)‘6(V7)]
dWa 1
-I----5v - -(M-5H' + Н'-Ш) - H0-5M}dr
dv 2
К8(п*у)п* dvdS.
(4-1)
This expression must be transformed to one containing only the
components of 3M or Sv and not such quantities as 3(Va) and 5H'.
To get rid of 3(Va) = V(3a), we use the transformation (3-50);
to get rid of 3H', we use the magnetostatic theorem (3-48). Then
(4-1) becomes
— *5vdr
(4-3)
where
H = Ho + H'
is the total magnetizing force, due partly to external sources and
partly to the poles in and on the specimen. The part H' is not
known in advance but must be found from M by the methods of
potential theory.
Equation (4-2) is of the form
SG =
P • 5vdr
(4-4)
To proceed further, we use method (a) of Section 3.6 and set
Sv = 30 X v, (4-5)
where 30 is a small vector rotation; this insures satisfaction of the
constraint v2 = 1 or v • Sv = 0. Equation (4-4) may then be written
dG = - fv X P-30dr - fv X Pa*80dS. (4-6)
For equilibrium, 3G must vanish for arbitrary S0. (It is not neces-
sary to require that 50 be perpendicular to v, since a component of
30 in the direction of v makes no contribution to v X P*30 or
v X Pe* 30.) Therefore v X P = 0 at each point of r, and v X P< =
0 at each point of S. Although each of these vector equations is
equivalent to three scalar equations in Cartesian coordinates, the
three are not independent, since the vector equation places a re-
striction only on the components of P perpendicular to v and not
on its component parallel to v. In fact, from @PZ — yPv = 0 and
yPx — aP2 == 0 we can derive aPy — ffPx = 0 by elimination of
Px. Thus in manipulating these torque equations, one must watch
for redundancies. Written in full, the equations are
dwa
= 0
dv
dv
dn
on S.
= 0
Equations (4-7) and (4-8) do not specify this boundary-value
problem completely. To complete the specification, we must relate
H' in equation (4-3) to its sources — V • M = — M8 V • v and n • M =
Meii*v. One method of doing so is to express H' or its potential U
directly by means of volume and surface integrals over the sources.
A usually more convenient method is to use the partial differential
equations and boundary conditions of potential theory. Thus
H = Ho - VU in r, (4-9)
II = Hq — V[7' outside r, (4-10)
where
V2U = 4тгМвГ-у, (4-11)
V2U' = 0,
and on S
U - U' = 0,
dU dU'
1- 4rMevn =
dn----------------dn
(4-12)
(4-13)
(4-14)
Equations (4-7) to (4-14), plus regularity of U' at infinity, specify
the equilibrium problem completely. It is obviously a formidable
one.
From equation (4-7) we can obtain an expression for the effective
field JC (cf. Sect. 1.3):
M8K = M8H + CV2v---------------- P. (4-15)
dv
The introduction of JC in no way helps in the mathematical solution
of the problem; it sometimes helps, however, in thinking about it
physically. (It may also hinder!) The surface analog of JC is 3C8
(dimensions H X length):
dv
M83C8 = — C-----K8ifvn. = Pe.
dn
(4-16)
As JC is the generalized vector force corresponding to generalized
vector coordinate M, so P is the force corresponding to coordinate
v.
The equilibrium equations (4-7) and (4-8) state that the vectors
P and Pe, or the effective fields, must be parallel to v; and these
same conditions can be written in the form (3-9), which one gets
directly by using the Lagrangian-multiplier method of imposing
the constraint M2 = M82 [method (6) of Sect. 3.6].
4.2 Static Stability Conditions
The stability problem is a much more complicated one. We may
take care of the constraint M2 = M 2 by using as dependent varia-
bles either the direction angles Ф and 0 or two direction cosines a
and /3 (then 72 = 1 — a2 — 02). For definiteness consider the latter
method. A small variation from an initial state (a0, /So) is described
by
a == «о + eUt 0 = 0o + 72 = 1 — (ao + ^)2 ~ (0o + ^)2.
(4-17)
We must now expand the variation dG to the second order in e.
For equilibrium, the coefficient of e must vanish for arbitrary
u(x, y, z) and v(x, y, z); this leads to equations equivalent to those
derived in the preceding section. For stability, the coefficient of e2
must be positive for arbitrary и and v (other than и = v — 0 every-
where). In most cases the expression for the term in G proportional
to c2 (the second variation of G) is very complicated, and the prob-
lem of demonstrating stability is a difficult one. The demonstration
of instability, however, requires only the discovery of one particular
variation u, v for which the second variation of G is negative.
One necessary condition for a minimum, the Lagrange condition
of the calculus of variations, is always satisfied in our problem. This
is the condition that the quadratic form containing the derivatives
da/dx etc., in the integrand of G, shall be positive definite. This
quadratic form is the result of expressing |C[(Va)2 + (V/3)2 +
(V7)2] in terms of Ф and 6 or of a and 3, a transformation that can-
not affect its property of being positive for any nonuniform
therefore the Lagrange condition is satisfied.*
When the applied field Ho is uniform, a necessary condition for
stability is that the reversible magnetization curve through the
equilibrium point considered shall have a positive slope at that
point. By the magnetization curve we mean a plot of the com-
ponent of volume-average magnetization in the direction of H0)
Мц, against Ho- If we let m = тМд, the component of the total
moment m = jMdr along Ho, we can prove the theorem as fol-
lows. Since G = A — mH0, the stable equilibrium states at speci-
fied Ho can be found by first minimizing A with respect to the func-
tions a and P at specified m, then minimizing G*(m) = A*(m) —
mH0 [where A*(m) is the result of the first step] with respect to m.
In the second step, the equilibrium <
dA *(m)
----------H
dm
and the stability condition is
d2A*(m)
dm2
is
(4-19)
0 ~ 0,
0.
(4-20)
* In terms of Ф and 0, this expression becomes
|C[(VO)2 + sin2e-(V$)2],
(4-18)
which of course vanishes for 0 = 0 or т and VO = 0, regardless of V$. The
basic difficulty here is that for О = 0 or т, Ф is indeterminate—a fact that at
times forces a change to a new polar axis or to a and 0 as variables. On the
other hand, the use of a and /3 sometimes causes difficulty at the points a2 4- 02
The reversible magnetization curve through a point (Яо, m) satis-
fying (4-19) is the locus of points also satisfying this equation and
located near the point under consideration; its inverse slope is
therefore
dH0 _ d2A*(m)
dm dm2
(4-21)
which must be positive for stability.
The stability problem becomes reasonably tractable when the
equilibrium state to be tested is one of uniform magnetization in a
uniform applied field. This case will be treated in Sections 5.2-5.4.
A simplified form of the stability problem occurs when the distribu-
tion is constrained to be a one-dimensional one in Cartesian or
cylindrical coordinates. These cases will be discussed in Chapter 6.
More general treatments of stability remain for future study.
4.3 Dynamic Torque Equation
The dynamic equation
dM
— = 7oM X JC,
dt
(4-22)
which is (3-5) with L written in the form (3-6), can be written out
in full by merely inserting JC from (4-15). The boundary equation
(4-8) remains unchanged. To include a dissipative torque by Gil-
bert’s method, we simply add to JC a term —[eq. (3-58)]. This
method of deriving the equation of motion is satisfactory for most
purposes.
Alternatively, we can use the variational principle of Section 3.3,
extended to the dissipative case as in Section 3.7, to derive the same
result. The essential steps in this derivation have already been out-
lined in those sections. Since 3 and $ make no contribution to the
surface integral, the surface equation remains the same as in
equilibrium.
We shall now prove the theorem mentioned in Section 3.3: that
at constant Ho, G remains constant in reversible motion and de-
creases in dissipative. By carrying out for actual changes in time
the same procedures used previously to derive equation (4-2), we
find
dG r dM r dv
— = — I jc----------dr = - I • — dr, (4-23)
dt J dt J dt
since the coefficient of in the surface integral vanishes in an
actual motion just as in equilibrium. By the equation of motion
(3-62),
d 53 53 5ff
MSK =---------------+ — + X'M, (4-24)
dt 5v 5v 5v
where X' is a scalar quantity. Since M • (dv/df) = 0,
(4-25)
Now 3 is homogeneous of degree 1, and homogeneous of degree 2,
in a and p or in a, /?, and у (whichever way the functions are ex-
pressed) ; therefore by Euler’s theorem on homogeneous functions
(Courant, 1936, pp. 108-110), v-(53/5v) = 3 and v-(5$75v) = 2£F;
also v*(53/5v) + v*(53/5v) = dZ/dt. Thus the terms containing
3 cancel (in a conventional nongyroscopic dynamical system, where
3 is of degree 2, they would give d3/dt\ and the right member of
(4-25) reduces to 2£E Hence
(4-26)
We have assumed that H' is related to M by the equations of
magnetostatics. At microwave frequencies this is a poor assumption
unless the specimen is very small; and even at much lower fre-
quencies it is unjustified if the conductivity of the material is large.
In such cases Maxwell’s equations can be used to relate H' to M.
The effects of finite propagation speed and of eddy currents are
then taken into account, and the energy relations are naturally
more complicated.
4.4 Steady-State and Transient Problems
Among the dynamic problems of special interest are (d) the
steady-state response to an alternating field and (b) the behavior
after a sudden change of the applied field. When the equations of
motion are linear, both these cases can be handled by standard
techniques such as Fourier analysis and the Laplace transforma-
tion. Such techniques transform from the time domain to the fre-
quency (real or complex) domain; the boundary-value problem in
space remains formidable except in special cases. Fortunately the
steady-state phenomena of interest are chiefly ones in which the
alternating field (because of power limitations at radio and micro-
wave frequencies) is perforce small; the equations can then be
linearized. In contrast, the transient problems of greatest interest
involve not small changes but large ones, such as changes from
positive to negative magnetization in a computer memory element;
and here the techniques associated with linear equations are useless
except for study of the initial phases of a reversal process, when the
changes are still small.
In principle a small oscillation can occur about any equilibrium
state, and any such oscillation can be studied by use of linearized
equations of motion. In practice, the small alternating field is
usually superposed on a large constant field; its function is to make
the equilibrium state, about which the oscillations occur, one of
uniform or nearly uniform magnetization. Occasionally the role of
the constant field is taken over by a large anisotropy that keeps the
magnetization uniform (after proper magnetic preparation). In
either case, the equilibrium state may be taken to be one of uniform
magnetization. This is fortunate from the theoretical point of view,
for it is only in this case that we can describe the equilibrium state
precisely and formulate precisely the equations that govern small
dynamic departures from it. In fact all theoretical calculations of
oscillations about other equilibrium states have been based on very
crude “domain” models.
The “small oscillations” mentioned are in general precessions,
but precessions of a complicated type: the tip of the magnetization
vector at each point of the body describes an ellipse rather than a
circle, and there are phase differences between different points.
When there is no dissipation, a free precession, which is along a
locus of constant G in the space of the functions a(r) and 0(r), can
occur about either a maximum or a minimum of G; there is no ex-
ponentially increasing departure from the maximum, as there is in a
conventional dynamical system with a kinetic energy quadratic
in the velocities. With dissipation, however, a precession about a
minimum decays toward it, whereas a precession about a maximum
decays away from it. A formal calculation of resonance conditions
with dissipation absent will yield both types of precession; inclu-
sion of a small dissipation term will determine which of the cal-
culated resonances represent precession about a maximum and are
consequently of no physical interest.
Although a complete study of transient processes by means of
linearized equations is not possible, their use to study the initial
stage of a reversal has the advantage that it can determine which
modes of deviation from the initial state build up fastest. Such
information can then be used to decide what modes to assume in
the nonlinear calculation. It appears that the possibilities of this
approach to the transient problem have not been exploited.
4.5 Dynamic Criterion for Static Stability
Instead of testing the stability of an equilibrium state M =
by examining the second variation of G, one can do so by the
following physically more illuminating method. Set up the lin-
earized equations of small motion about the equilibrium state, with
dissipation included, and examine the transient behavior after a
small initial deviation is introduced. The modes of natural motion
will be of the form
a(x, y, Z, t) = a0(x, y, z)e~pi, (4-27)
fi(x, y, z, t) = p0(x, y, z)eTpt, (4-28)
where p is in general complex. For stability, the real part of p must
be positive for every natural mode.
This method is of course equivalent to the more formal one dis-
cussed earlier. It has the advantage that the necessary mathemati-
cal steps may become more obvious if the problem has been formu-
lated in this physically meaningful form.
For merely testing the stability, it is not necessary that the as-
sumed law of motion be the actual one, provided the potential
energy G has the correct form. Therefore the kinetic function 3
can be omitted and only the dissipation function $ retained. The
assumed motion is then one of simple viscous decay, either toward
a minimum or away from a maximum.
In classical dynamics, stability is often tested without introduc-
tion of dissipation, by examining whether the natural motions are
oscillatory or aperiodic. That method will not work here if the
actual equation of motion is used, for the motion about either a
minimum or a maximum is periodic. The method becomes applica-
ble if the gyroscopic kinetic term in the Lagrangian is replaced by a
conventional “kinetic energy” quadratic in the velocities. However,
the use of a dissipation function gives equivalent results in a way
that is both more realistic and intuitively more enlightening.
4.6 Linear Approximation
We saw in Section 4.4 that small precessions about an equilib-
rium state of uniform magnetization are important experimentally
and can be studied by use of linearized equations. There are also
equilibrium problems of linearizable type: namely, those in which
a large field is present, so that the deviations from a reference state
of uniform magnetization are small. We can derive the basic equa-
tions for both these cases simultaneously.
We take as our reference state one of uniform magnetization.
Such a state can be maintained as an equilibrium state if the speci-
men is a homogeneous ellipsoid, with any relative orientation of
the crystal and ellipsoid axes, and with any direction of the mag-
netization with respect to the ellipsoid axes. In this case the main-
tenance of the uniform state will in general require components of a
uniform applied field both transverse and parallel to the magnetiza-
tion. The transverse components are necessary to equilibrate ani-
sotropy forces and demagnetizing fields, the longitudinal compo-
nent to stabilize the state; for it is clear without detailed mathema-
tical analysis that a very large field along +M will cause the
external-field energy — jM*Hodr to have a sharp minimum in this
direction, whereas a very large field in the opposite direction will
cause it to have a sharp maximum. We choose the direction of the
uniform magnetization in the reference state as the z axis (Fig.
4.1). In general this will not be the direction of the applied field
Figure 4.1. The reference state for the linearized micromagnetic equations.
Ho(o) in the reference state. We shall usually assume that H0(0)
(though not necessarily Ho) is uniform.
We now suppose that the magnetization deviates from the
reference direction because of a failure to satisfy the equilibrium
conditions in the reference state exactly, but that the deviations
are small enough to justify in G neglect of terms of third and higher
orders in a, 3, and quantities that vanish with them. This means
that G} 3, and 5 are to be expressed to the second order in these
small quantities and that the equations of motion or of equilib-
rium will be linear in them. To the second order, with zeroth-order
terms omitted,
G = f |J C[(Va)2 + (V/3)2] + g,a + grf
+ - fall®2 + + g22&2]
In (4-29), the method of equation (3-49) has been used to simplify
the magnetostatic term; II'(0) = Я'х(0)1 -|-is the field of the
poles produced by the uniform magnetization in the reference
state,* whereas H/(1) — H*+ • • • is the field of the additional
poles produced by the small transverse magnetization M8(ai +
/?j) = M8u. The first volume integral extends over the specimen,
the second over all space; in the surface integral, (Z, m, n) are the
* The superscript (0\ indicating “reference state” (i.e., of zeroth order in a
and 3), must not be confused with the subscript o, indicating “applied” (i.e.,
produced directly by external sources).
direction cosines of the outward normal n. The coefficients gi and
gij are found by expanding the anisotropy-energy density to the
second order in a and 3- A time derivative has been dropped from 3
(cf. Sect. 3.3).
If the reference state is to be an equilibrium state, the first-order
terms in G must vanish, for arbitrary a and /3, when Ho = Ho(o).
This requires
gi - M8(HQx^ + H'x™)
= g2- M8(H<^ + Я'/0)) =0 in т, (4-32)
K8ln = Ksmn = 0 on &
For a homogeneous single-crystal ellipsoid of arbitrary shape and
orientation, the volume conditions can be satisfied by adjustment
of the transverse components of the applied field. If the specimen is
not an ellipsoid or not homogeneous, the conditions will not usually
be satisfied, because in the first case H'x^ and Я'у(0) are not uni-
form, and in the second the g/s are not. We therefore separate Qi
into a term gf® that satisfies (4-32) and a term g^ whose effect
on a and /3, by hypothesis, is small; and we define the reference
state as one in which the gfs have the values g^\ The reference
state is then one of equilibrium as far as the volume conditions are
concerned.
The surface equilibrium conditions (4-33) cannot be satisfied,
except for special geometries, unless K8 — 0. We therefore com-
plete the definition of the reference state by specifying that in it,
K8 = 0. Then if surface anisotropy is in fact present, we shall re-
gard it as a first-order perturbing effect, along with g^l}. This is
legitimate, since the surface anisotropy is small.
Finally, we separate the applied field Ho into its value H0(0) =
H---------in the reference state and a part ho = hQxi 4-------
that acts as a perturbing field, e.g., an alternating part. Thus we
have three causes of departure from the reference state: the
K8f and ho; and in the reference state, K8 = 0 and
Pi(0) - Ms(HQx^ + H'x™) = (72<O) - + Я'/0)) = 0.
(4-34)
We assume that deviations of the 0</s from uniformity anp them-
selves first-order small quantities and therefore produce only third-
order terms in G, and that the term ^M9hQZ(a2 + 02) is also U third-
order small quantity. (Note that we have already omitted the
zeroth-order term — M8h^ which, though perhaps time-dependent,
is independent of a and /3 and therefore does not affect the motion.)
In the surface anisotropy, however, we tentatively retain the terms
quadratic in a and /3.
With these assumptions, we get
+ 2lma0 + (m2 — n2)02] dS.
(4-35)
We can now carry out a standard variational procedure and
equate the coefficients of da and 80 to zero, since in this сане a and
/3 are subject to no constraint. The result is: in r,
O dU 9 M9
CV2a — (к + дц)а — gi20 — M8---------M2T)a--------0
дх To
= - M8h^ (4-36)
9 dU 9 . M8
CV20 — 012a — (к + g22)0 — -----M2q0 H-------a
ty Vo
on/8, = P2(1) (4-37)
da ft
C — + - n2)a + Imp] = ~K,ln, (4-38)
dn
«Э/3
C — + K„[mla + (m2 - n2)fl = -K,mn. (4-39)
dn
Неге к = M8[Hqz^ + Я'/0)Ъ and is the magnetostatic poten-
tial due to the small transverse magnetization Me(ai + /3j). The
equations have been written wTith the terms of first degree in a and
3 on the left and the terms of zero degree on the right. Since the
equations are linear, the effects of the internal irregularities
ef the surface anisotropy terms on the right of (4-38), and of the
time-varying field h0 can be found separately and superposed.
The equations of motion (4-36) and (4-37) can also be obtained
“rom the general equations of motion by expansion to the first
>rder of small quantities.
Chapter 5
Static Problems: Linearizable Cases
5.1 Approach to Saturation
The first calculations in micromagnetics were the one-dimen-
sional ones of Landau and Lifshitz and of Bitter; and the former of
these, the “wall” calculation, contains essentially all the micromag-
netics ordinarily used in domain theory. A complete and self-
consistent theory, however, must be three-dimensional. A direct
attack on the three-dimensional problem in all its generality leads
to nonlinear partial differential equations (Sect. 4.1) for which no
analytical solution techniques exist. Under such conditions it is
natural to look for cases in which the equations can be linearized;
the solutions in these cases not only have value in themselves but
may be useful as starting points in the attack on nonlinear cases.
In the equilibrium problem, linearization becomes possible when
the magnetization direction is almost uniform, a condition usually
attainable only by the application of a large uniform field to a
specimen of suitable geometry. The linearized equations for this
case were derived in Section 4.6 and are equations (4-36)-(4-39)
with h0 = 0, a = /3 = 0: in t,
CV2a — (к + 0n)a — — Ms— = £i(1), (5-1)
ox
dU
cv2p - p12a - (к + g22)fi - Me — = S2(1); (5-2)
dy
62
micromagnetics
on S,
(5-3)
C----h K8[mla + (m2 — n2)fi] = — K8mn.
dn
To review the notation: the spontaneous magnetization M8 is
directed almost along the z axis, with small direction cosines a, 0
along the x and у axes; к = M8[HQgw + = M8HZ{Q\
where H/0) is the z component of the magnetizing force in the
reference state of uniform magnetization along Oz\ the are the
coefficients of the quadratic terms in a and 0 in the Taylor’s ex-
pansion of the crystalline-anisotropy energy density about the ref-
erence state, and they are assumed to be independent of position;
the £i(1),s are the parts of the coefficients of the linear terms that
are not equilibrated by the transverse components HX(Q\ Hy(0) in
the reference state, and they may vary with position; C is the ex-
change constant, K8 is the surface-anisotropy constant, and
(Z, m, ri) are the direction cosines of the outward normal n. The
magnetizing force H(0) in the reference state is the sum of the ap-
plied field intensity H0(0) and the field intensity H'(0) of the poles;
if the specimen is an ellipsoid (in any orientation) and if Ho(o) (as
will be assumed) is uniform, then H'(0) is also uniform and is ex-
pressible by means of demagnetizing factors. Since even the linear
equations become rather unmanageable when к is spatially variable,
we shall usually assume that the conditions for uniformity of H(0)
are satisfied. The potential U is that due to magnetization Ms(ai +
(3j) and is determined by
9 /da d0\
V2U = 47tMJ — + —) in t, (5-4)
\dx dy/
V2U' = 0 outside r, (5-5)
dU dU'
U = Uf and-----------H 47rMe(Za + mff) —-------on S,
dn dn
(5-6)
together with regularity at infinity.
Even the linearized problem is evidently complicated. Neverthe-
less, solutions of several important problems have been obtained
by this method.
The effect of random deviating forces — <7i(1), —02(1) has been
calculated (Brown, 1940b) on the assumption that the дц terms are
negligible, that the random sources are sets of statistically inde-
pendent and uniformly distributed points, lines, or planes, and that
the specimen dimensions are large in comparison with the distance
over which the effect of a local disturbance becomes negligible; the
specimen may then be regarded as infinite (once its shape has been
taken into account in the evaluation of H'(0)), and boundary condi-
tions can be ignored. Under the conditions stated, equations (5-1)
and (5-2) can be written in vector form (u = ai + 0j)
к X [CV2u - ku - MSVU - g(1)] = 0, (5-7)
where g(1) = When there is no variation with z
(e.g. in the case of infinite line sources along Ог), the quantity in
brackets itself vanishes; then V X u attenuates with distance from
a source approximately as e~Xir, and V-u approximately as
e~X2r, where
X/ = K/c = (5-8)
X22 = (к + 4tM?)/C = M8BZW/C, (5-9)
with
B,(o) = Hz(0) + 4ttM8. (5-10)
In the general three-dimensional case the mode of decay is more
complicated; but when H2(0) »4тгЛ/8, the term M8VU in the
brackets in (5-7) can be neglected, and u attenuates as e~Xir or
e—Aar (these then are equivalent). The treatment of the specimen
as infinite is justified if its dimensions are large in comparison with
i/Xi.
The high-field behavior shows no tendency toward domain for-
mation: the effect of a local disturbance simply dies off with dis-
tance from the source. If, on the other hand, we retain the дц
terms and suppose that we can continue to use the linearized equa-
tions down to fields such that к + g# changes sign, then we get, not
an exponential decrease with distance, but an oscillation. As we
shall see in Section 5.4, this general method can be used in some
cases to draw conclusions about the form of an incipient domain
structure.
Explicit solution of equations (5-l)-(5-2) and (5-4)-(5-5) for a
and 0 in finite form is not usually possible; but fortunately this
is not necessary if all that is desired is the value of SI h, the com-
ponent of mean magnetization in the field direction. To the sec-
ond order of small quantities,
= -MsJ’ydr = Ms 1 - 4 J*(«2 + 02)<fr}. (5-11)
If 0i(1) and are expressed as Fourier integrals in x. y, and z,
the Fourier transforms of a and 3 can be found without difficulty;
and these transforms can be used directly to evaluate, by use of
ParsevaFs theorem (Jeffreys and Jeffreys, 1956, pp. 457-458), the
integral f(a2 + ff2)dr and hence Mjj.
For random deviating forces, in the range H s Нг^} 4тгЛ/5,
point, line, and plane concentrations of the sources lead to formulas
of the form
Stn/M9 - 1 - a/HnZ2, (5-12)
with n = 1,2, and 3 respectively in the three cases. Forces uniform
throughout an extended volume give (Becker and Boring, 1939,
pp. 168-171) a similar law with n = 4. Experimental data have
often been fitted to an empirical function of the form 1 — a/H —
b/H2. The presence of the term —a/H suggests that besides the
extended forces commonly assumed, and responsible for the
~b/H2 term, there may be line concentrations. A more realistic
model, however, must replace this rather formal concept by the
“dislocations” that have proved useful in explaining the plasticity
and other properties of metals. Line dislocations are, from the mag-
netic point of view, not line concentrations of force; for the mag-
netic effects are produced by the stress field of the dislocation,
which is not concentrated at the dislocation itself. A calculation
based directly on the dislocation concept (Brown, 1941) led to
much more complicated results. This work has recently been ex-
tended (Seeger and Kronmtlller, 1960, 1961) and improved by
introduction of more modern ideas about dislocations. Unfor-
tunately the formulas are extremely complicated. It is to be hoped
that this approach will ultimately lead to a picture of “internal
stress,” as it shows up in magnetic phenomena, more realistic than
the very vague picture that prevailed in the 1930’s.
The main cause of the complexity of these calculations is the
presence of the magnetostatic term —MgVU in equation (5-7).
If this term is neglected, a simple direct solution is often possible;
but such neglect is justified only when H » 4tM8, a condition
seldom satisfied. The Becker-Kersten “internal stress” theory of
the coercivity neglected magnetostatic effects even at zero H, an
unjustified simplification that ultimately forced the virtual aban-
donment of the theory. Domain theory of the Landau-Lifshitz
type goes to the opposite extreme and assumes that the magneto-
static forces are powerful enough to eliminate poles completely.
The linearized high-field equations of micromagnetics have the
virtue that they permit a rigorous calculation of magnetostatic
effects, though sometimes only by enormous labor.
Magnetostatic effects similarly complicate the approach to
saturation of a soft polycrystalline specimen. For H 4тМв and
negligible exchange effects, an elementary calculation gives a 1 —
b/H2 law for Мн/М8. This is easily seen from equation (5-7);
for with the C and M8 terms omitted, this equation gives simply
u = — g(1)A and hence, by equation (5-11),
-1 - A
~M~8
2M„2H2 ^Av’
(5-13)
where (g<1)2hv = <Si)2)av + (<92)2)avis the volume average of the
sum of the squares of the coefficients of a and Д in the anisotropy-
energy density. These coefficients vary from one crystallite to
another, but the volume average is easily calculated for an isotropic
random distribution of orientations of the crystal axes. If now the
magnetostatic term is included, the problem is far more compli-
cated; but the Fourier method can again be used. This calculation
was carried out by Holstein and Primakoff (1941) and repeated by
N£el (1945, 1948b). The effects of the exchange term CV2u and of
the second-order anisotropy coefficients дц were neglected. The
result is that for H « 4тгЛ/8, there is a 1 — bf /Н2 law, with bf =
for H ~ 4irM8, the behavior is more complicated. This curious
phenomenon, formulas of the same form but with different coeffi-
cients for the ranges H » 4тгМ8 and H « 4?rMe, was encountered
also in the case of concentrated forces; but the formulas for H «
4тгМ8 are of small importance, for in this range the calculated a
and /3 are usually too big for applicability of the linear theory.
Neel (1948a) calculated the effect of cavities on the approach to
saturation. His method was first to assume small deviations of Ms
from its average value* and then to apply the results of this
linearized theory to the case of cavities; the second step is obviously
open to criticism. N6el interpreted the empirical 1/H term as an
approximation to his formula, and he successfully fitted experi-
mental data on porous iron. He also criticized the dislocation in-
terpretation of the 1 /Н term, but his criticisms seem to have been
based on a misunderstanding (Brown, 1951a); whatever the short-
comings of the formal dislocation calculation, it seems clear physi-
cally that effects dependent on plastic flow must be attributed to
dislocations and not to cavities.
Results of the linearized calculations have been used to evaluate
quantities needed in domain theory: by Ndel (1946) to evaluate
magnetostatic energies associated with domain wall configurations,
and by Vicena (1955) for similar evaluations of wall energies as-
sociated with dislocations.
5.2 The Coercivity Paradox; the Role of Imperfections
In the Akulov rotation theory (Sect. 2.1), it was assumed that the
vector magnetization of a crystal remains uniform as it rotates. If,
then, the magnetization and the field are initially along a “direc-
*This requires some modification of equations (5-l)-(5-6).
tion of easy magnetization” (minimum of anisotropy energy) 0г,
which is also a principal axis of the ellipsoidal specimen, and if the
applied field is now decreased to negative values, no change should
occur until the initial state becomes unstable with respect to a uni-
form deviation a = «о, 0 = 0o- Such a deviation involves no change
of exchange energy; and except in an extremely fine particle (of
diameter ^102A), the surface energy will be negligible in compari-
son with the volume energy. The magnetostatic self-energy is
лт = &M2{Nxa2 + Nyffi + yz72}
(5-14)
where NXi Ny, and Nz are the demagnetizing factors of the ellipsoid;
the change of this from the reference state a = /? = 0, 7 = 1 to
the state a = 0 = /30, T2 = 1 — («о2 + 3o2) *s
SAm = (Nx - + (Nv - Уг)3о2}- (5-15)
Therefore the net change of free energy per unit volume, to the
second order of small quantities, is
bG = К^цло2 + + 022/?o2) + lAf3Ho(ao2 + /tf)
+ i<32{(Nx - NJaJ + (Ny - NZ)0Q2} (5-16)
and is positive as long as the quadratic form with matrix
~gil + M2(NX - NJ + MSHQ g12
-gi2 922 + M2(Ny - NJ + МвЯ0-
(5-17)
is positive definite. For an ellipsoid of revolution about Oz, Nx =
Ny = Nb> Nz = Na] and by rotation of axes about Oz we can make
<712 = 0. Then instability occurs when M8H$ becomes equal to the
negative of the smaller of the two quantities да + M2(Nb — NJ;
that is, when M8H = M8(Hq — УаМ8) becomes equal to the nega-
tive of the smaller of the two quantities дц + M2N^
For a cubic crystal with anisotropy energy density Ki(a2ft2 +
/3272 + 72«2) and with > 0, the direction [001] is an easy direc-
tion; for small deviations from this direction, the anisotropy-energy
density to the second order is Кг(а2 + /32), so that gn = g22 =
2Klt and the critical magnetizing force for instability is
[2Лл
Hc= - — + MsNb\.
I M,
(5-18)
For iron at room temperature, M, ~ 1.7 X 103 emu, Kx ~ 4.3 X
105 erg/cm3 (Bozorth, 1951, pp. 567, 871). Since Nb > 0,
2ZC
\He\> —- « 5 X 102oe.
(5-19)
Ms
In experimental magnetization curves on iron crystals at or near
orientation [100] (Bozorth, 1951, p. 561), the magnetization M
(plotted against H = Hq — NaM) remains constant as H de-
creases, until H reaches a certain negative value —Я'с; M then
decreases very rapidly to negative values, along a curve that
is almost vertical, so that the coercivity is practically equal to 7Z'C.
The value of H'Cf according to the theory just described,
should be about 5 X 102 oe; experimentally, it is about 0.1 oe.
Thus the rotation model cannot be correct at small H. This fact
was recognized very early and was the reason for the introduction
of the “inversion” concept.
When “inversion” was interpreted as “wall displacement,” the
low H'c and resulting low coercivity were considered explained. In
an ideal crystal the position of a wall, according to the Bloch (1932)
or Landau-Lifshitz (1935) calculation, does not affect its energy;
and if we modify the theory by letting the anisotropy constant vary
slightly with position along the direction normal to the wall, we
find that if the variation is slight, a small field can displace the wall
a large distance. Thus wall displacement is a low-energy, low-field
process, limited only by small deviations from ideality; and in the
presence of walls, the coercivity should indeed be low.
But for success of this explanation, the walls must first be
present. And if we do a rigorous three-dimensional calculation for
an ellipsoidal crystal, initially uniformly magnetized by a large ap-
plied field, we find that walls do not form within the field range in
which they must be present if they are to explain the observed mag-
netization curves. We can see this as follows (Brown, 1945).
As usual, we take the direction of the original uniform magneti-
zation as the z axis. The state to be tested for stability is then the
state a = ft = 0. For small deviations a = eu, 0 = tv from this
state, equation (3-44) gives, to the second order in e, i.e., in a and 0,
+ 1 М„Нгт(а2 + 02)} dr + — Гн/(1)2</т; (5-20)
2 8т J
this is equation (4-35) with the linear terms omitted, since they
must be zero if state a — ft = 0 is one of equilibrium. (This im-
plies absence of surface anisotropy, of which we shall say more
later, and equilibration of any transverse anisotropy forces by
transverse fields.) From equation (5-20) it follows that positive
definiteness of the quadratic form with matrix
Г<7и + M8H™
912
L012
g22 + MSHZW J
(5-21)
is a sufficient condition for stability, not merely with respect to a
uniform deviation a = a0, 0 = but with respect to an arbitrary
deviation a(xf y, z), ft(x} y, z). For the additional terms containing
C and are nonnegative; in fact the former is positive for any
nonuniform a and 3, the latter for any uniform a and 0.
It follows that in an ideal crystal, no deviations from the initial
state, such as are necessary for the formation of walls, can occur
before H = has reached the larger of the two values found by
setting the determinant (5-21) equal to zero. By rotation of the
x and у axes we can make ^12 = 0 and > 922} then the value of
H above which no deviation from saturation should occur is
—011/^G- For the iron crystals discussed in connection with equa-
tion (5-19), 9i\/M& = 2K\/MS « 5 X 102 oe. Thus consideration
of all possible modes of deviation from saturation, rather than
merely of a uniform rotation of the magnetisation (“rotation in
unison”), has replaced the term M9Nb in equation (5-18) by an
unknown nonnegative quantity but has left the inequality (5-19)
unaffected; there is still a serious discrepancy with the experimental
curves, which show a drastic deviation from saturation at H
~ —0.1.
This “paradox” is best stated in terms of the value of H (or of
Ho = H + NaM9) at which a deviation from saturation first oc-
curs. It was originally stated (Brown, 1945) in terms of the coerciv-
ity; but the predictions of the theory with respect to coercivity are
indirect and rather complicated. For a specimen with Na = 0,
dH/dM = dHo/dM is nonnegative; for it is positive in reversible
parts of the curve, by equation (4-21), and is zero in irreversible
jumps. Therefore H, having reached the value —0ц/ЛГ8, cannot
increase as M decreases to zero, and the theoretical prediction is
that the coercivity should exceed <7п/7И8.* For a specimen with
Na > 0, however, the quantity that must be nonnegative is not
dH/dM but dHo/dM; and an irreversible jump to M < 0 at HQ =
const. = NaMg — ди/Ms — e (where e is some small positive
quantity) would give a coercivity gu/M9 — NaM9 + e. With
large Na, this may be considerably less than дц/М9.
Spatial variations of the cannot remove the discrepancy
between theory and experiment unless the minimum дц/М9 is of
order 0.1 oersted, i.e., 50г00 of its normal value; for positive def-
initeness of the quadratic form (5-21) al every point of the crystal
is still a sufficient condition for stability of the original uniformly
magnetized state.
Thus the success of the wall theory in explaining the observed
* Until the magnetic properties of films and fine particles were discovered,
it was commonly assumed that once the M vs Hq curve had been “sheared”
to obtain the M vs H curve, all effects of specimen geometry (at least in the
case of an ellipsoid) had been removed; in other words, that the M vs H curve
was a property of the material, independent of specimen geometry. We know
now that this is not true. Empirically, however, it seems to be at least approxi-
mately true of ordinary bulk material. Why this should be so, and under what
conditions, is an unsolved theoretical problem.
low-coercivity magnetization curves was a delusion; the argument
was in fact circular and collapses under rigorous scrutiny. The wall
concept itself is not sufficient to explain the low-field behavior; we
need also a mechanism for the nucleation of at least a few isolated
domains, and with them of walls.
There can be little doubt that what is needed for such nucleation,
and is in fact usually present, is imperfections of some type (Stoner,
1950, p. 116). Various attempts have been made to make this hy-
pothesis precise (Brown, 1959a; Aharoni, 1960,1961; Abraham and
Aharoni, 1960; Shtrikman and Treves, 1960; Aharoni, 1962b);
but the difficulties are great, and no outstanding success has been
achieved in modifying the theory to fit the usual experimental
situation. There has been greater success in modifying the experi-
mental situation to fit the theory (DeBlois and Bean, 1959): se-
lected iron whiskers do show a persistence of saturation until a
field has been reached that is close to that predicted by the theory.
Quite possibly an essential condition for low-field nucleation of
domains is the presence of transverse deviating forces of appreciable
magnitude, as distinguished from mere fluctuations of the 0t/s.
Such transverse forces can arise from anisotropy fluctuations, from
dislocations, from magnetostatic fields at cavities or edges, or from
surface anisotropy. Attempts to take account of them by including
terms in g^ and in K9 in the linear theory lead only to the con-
clusion that by the time instability occurs, the linear theory is no
longer applicable (Brown, 1959a). Thus a completely satisfactory
resolution of the coercivity paradox apparently requires the carry-
ing out of a nonlinear calculation for some simplified but plausible
model.
The “nucleation” calculations of standard domain theory (e.g.,
Goodenough, 1954) evade the basic difficulty without resolving it:
they show only that in certain cases the final free energy will be
less if a domain can form. We must also show how to reach the
final state without intermediate increase of free energy. Thermal
agitation is not a sufficient explanation; for its role is primarily to
start a system away from an equilibrium state that has already be-
come practically unstable. Until — Я is comparable with 2K\/M8i
the energy barrier to be overcome is still of order 4 X 105 erg/cm3,*
and so the thermal energy fcT « 4 X 10”14 erg (at room tempera-
ture) can turn the magnetization into an unfavorable direction
only in a volume of order 10~19 cm3, e.g., in a cubical region whose
linear dimensions are comparable with the thickness of a wall. This
cannot, with significant probability, produce a reversed domain
surrounded by a wall. (See also Aharoni, 1962a.)
5.3 General Theory of the Nucleation Field
In view of the difficulty of taking account of the imperfections
that are apparently important in ordinary materials, further de-
velopment of the idealized theory of Section 5.2 is of interest only in
cases in which imperfections may be expected to play a minor role.
There are two such cases. One is the fine particle, for which there is
an appreciable probability of freedom from defects; the other is the
case in which the initial magnetization direction is one of difficult
magnetization, so that instability occurs at a positive field, large
enough to make the effect of imperfections still insignificant.
Let us assume that it is surface defects that are important, that
the probability that an infinitesimal surface element AS contains
one defect is ?A8, that the probability that it contains more than
one defect is (ХД82), and that different surface elements AS are
statistically independent with respect to defects. Then the number
of defects in a finite surface area S is a random variable with a
Poisson distribution: the probability that S contains n defects is
e~~vS{vS)n/n\y the mean number of defects in an area of the size of
S is vS, and the probability that it contains no defects is e~vS. The
whisker experiments, with a typical whisker 1 cm long and 10-3 cm
thick, show that this ceases to be a negligible probability when
S ~ 10~3 cm3. If for definiteness we say that c~vS = 10~3 when
8 = 4 X IO"3 cm, we find v = 1.7 X 103 cm’2. Then a particle
of surface area 10“8 cm2 (linear dimensions ~1 micron) has vS e
1.7 X 10~5 and e~*s = 1 — 1.7 X 10~5; that is, it is practically
* If exchange energy were included, this estimate might be even higher.
certain to be free of defects. Therefore the theory of ideal crystals
should be quite reliable for single-domain and almost single-domain
particles, whose dimensions are even smaller than 1 micron.
Returning to the situation examined in Section 5.2, we assume
for simplicity that H = is uniform. Let us suppose that H is
decreased below the value at which the quadratic form (5-21)
ceases to be positive definite. Instability will ultimately occur. The
critical value of H at which it occurs—the “nucleation field”—can
be found by the following reasoning.
For any assumed variation a — eu, @ = tv (with и and v arbi-
trary functions of x, y, z)f a value of H can be found for which bG =
0: namely, from equation (5-20),
-MSH = -» (5-22)
with ®
: | C[(Va)2 + (V0)2] + J [0п«2 + 2£712«3 + i72232]) dr
Q = f 1 (a2 + ^dT- (5-24)
</ 2
If H is decreased below this value, the reference state will become
unstable with respect to this particular и and v. We are interested in
finding the largest H at which instability can occur. We therefore
minimize P/Q, or P for given Q, with respect to the functions и and
v, or a and (3. The general theory of such a minimization problem
is discussed by Courant and Hilbert (1931), Chap. VI. Only an
elementary discussion will be attempted here.
To find a necessary condition for such a minimum, we set the
first variation of P/Q equal to zero. This requires QbP — PbQ = 0,
or
bQ - \ЬР = 0, (5-25)
where
P
X = - = -М,Я(= -к). (5-26)
Q
On carrying out the variational procedure, transforming the C term
in the usual way, and setting the coefficients of da and 6/3 equal to
zero, we get in т
O dV
-CV2a + (gn - X)a + д12/3 + M8----= 0, (5-27)
dx
о dU
— CV2/3 + 012a + (^22 — b)0 + M8 — = 0,
dy
and on S
да d{3
c— = c— = 0.
dn dn
(5-28)
(5-29)
Here U is the potential of i.e., H'(1) = — Vf7; in obtaining
the variation of the H'(1)2 term in P, the transformations of Section
3.5 are again useful.
These partial differential equations and boundary conditions,
together with those that relate U to its sources, describe a linear
boundary-value problem. Nonvanishing solutions exist only when
X has one or another of a set of characteristic values (eigenvalues).
If we can find the smallest eigenvalue Xo, we are assured that no
deviation can occur for M8H > — Xo; that is, —\Q/M8 is the
nucleation field (expressed as magnetizing force, not as applied
field). The corresponding solution a, /3 (indeterminate by an arbi-
trary multiplicative constant) gives the form of the initial devia-
tion that occurs when H reaches the nucleation value.
Equations (5-27)-(5-29) are simply equations (5-l)-(5-3) with
the deviating forces g^ and K8 omitted. The nucleation field is
the field at which a deviation without any deviating force first
becomes possible, or at which the equilibrium of the state a = /3 =
0 becomes neutral. The equations can be obtained more easily
from this alternative point of view; but the method given above
makes it easier to see how the calculated nucleation field will be
affected by the introduction or removal of constraints, a technique
very useful for approximate or preliminary calculations and for
quick rejection of higher eigenvalues.
5.4 Nucleation-Field Calculation in Specific Cases
To determine the nucleation field for a particular specimen
shape, it is necessary to find the (algebraically) smallest eigenvalue
Xo of the boundary-value problem (5-27)-(5-29). Each eigenvalue
Xn corresponds to a certain mode of deviation an, fin; which mode
has the smallest X depends on the size of the specimen. We shall
consider in detail only the prolate spheroid and its limiting cases,
the infinite cylinder and the sphere, with the applied field Ho and
the initial magnetization along the axis of revolution and with
0n = 022 0» 012 =* 0- Then equations (5-27) and (5-28) become
о 3U
-CV2a + (g - X)a + M8------= 0, (5-30)
dx
_ dU
-CV20 +(g-\)V + M8 — = 0. (5-31)
dy
One solution is a uniform deviation a = const, /3 = const, “rota-
tion in unison/’ This is the mechanism assumed in the Akulov rota-
tion theory and in the Stoner-Wohlfarth single-domain-particle
calculation. The potential problem can be solved immediately and
gives dU/dx = NbMsa, dU/dy = NbM8ft, where Nb is the trans-
verse demagnetizing factor. The terms V2a and V2/? are zero, and
equations (5-30) and (5-31) possess nonvanishing solutions if g —
X + NbM2 = 0. Since X = —M8H = —M8(HQ — NaM8), where
Na is the longitudinal demagnetizing factor, this gives
M8Hq = -[0 + M2(Nb - Na)], (5-32)
which is (except for a factor M8) the Stoner-Wohlfarth “critical
field” for a particle with the applied field along its axis. Equation
(5-32) can also be obtained in the manner discussed after equation
(5-16).
Another set of solutions, characterized by the property that u
has no volume or surface divergence, is described in cylindrical co-
ordinates (z> p, Ф) by
Иф = иф(2, p),
иг = up = 0.
(5-33)
Since there are no poles, €7 = 0; and on substituting a = — иф sin ф,
/3 = иф cos ф in (5-30) and (5-31), we find that иф must satisfy
with
(5-34)
(5-35)
and that диф/дп must vanish at the surface. For a cylinder of radius
6, the solutions are of the form
иф = BJi((k2 — cos (дг — 6), (5-36)
with д arbitrary and with к so chosen that
/i'((fc2 - aW) = 0. (5-37)
The smallest X is that corresponding to the smallest fc2, i.e., to the
smallest root x\ of = 0, approximately 1.84, with д = 0.
Then X = g + Cx\2/b2. For the cylinder, Na = 0, Nb = 2тг.
Hence for this mode, “magnetization curling/7 in a cylinder
M8H0 = -
Cx\2'
(5-38)
whereas for rotation in unison in a cylinder
M8H0 =-[{/ + 2ttM82].
(5-39)
If we assume tentatively that no other modes have smaller
eigenvalues, instability will first occur with respect to curling or to
rotation in unison, according as (5-38) or (5-39) gives the smaller
value of —M8Hq. The two are compared in Figure 5.1. In each
case there is a term g that represents the field necessary to over-
come crystalline anisotropy; this term is independent of the cyl-
inder radius b. For the curling mode, there is an additional term
that represents the field necessary to overcome exchange forces;
this term decreases with increasing b, because the angle between
successive spins decreases. For the rotation-in-unison mode, there
Figure 5.1. Analysis of the nucleation fields for two modes of deviation from
an original uniform longitudinal magnetization in an infinite cylinder. [After
Brown (1959b).]
is no exchange term, but there is a magnetostatic term representing
the field necessary to rotate in opposition to the transverse demag-
netizing field; this term is independent of b. The preferred mode is
rotation in unison at small b and curling at large 5. The change
from one mode to the other occurs at the critical b, bcy for which
the two curves intersect:
6e =
x'l
M,
(5-40)
These terms in the nucleation field receive a direct energy inter-
pretation in equation (5-22). This equation, applied in particular
to one of the modes of deviation under consideration, states that
at the nucleation field for this mode the decrease of energy, per
unit of |J(a2 + ffydr, due to rotation toward H must equal the
increase P/Q due to exchange, anisotropy, and magnetostatic
forces, so as to make the equilibrium at this H neutral.
4
It is now necessary to investigate the other solutions of the
boundary-value problem. For this purpose it is not necessary to
find every solution, but only to demonstrate that those not found
have larger eigenvalues than ones already found. A useful technique
for this demonstration is to drop from P a nonnegative term, such
as the magnetostatic term containing H,(1)2. The resulting problem
can often be solved more easily than the original one; and if it gives
for the mode under consideration (e.g., one for which up and иф
have period 2т/п in ф) a larger X than the X’s already evaluated,
the correct X for that mode is certainly too large to be of interest.
By such methods (Aharoni and Shtrikman, 1958), it has been
shown that for the cylinder there is just one mode that sometimes
has a smaller X than those already considered. It may be described
approximately as a rotation in unison at each cross section, with
a sinusoidal variation of amplitude along the cylinder axis; the
added exchange energy (due to the sinusoidal variation) may be
outweighed by a decrease of magnetostatic. The mode thus de-
scribed is called “magnetization buckling” ; the actual mode differs
slightly from it. As Figure 5.2 shows, the nucleation field for this
mode differs only slightly from that for rotation in unison over the
radius range where either requires consideration; for most purposes,
therefore, the simpler rotation in unison is a sufficient approxima-
tion.
We turn now to the sphere. Instead of (5-36) with g = 0, we get
иф = Ajnikr'jPn1 (cos в) (n > 1), (5-41)
where jn(x) is a spherical Bessel function in the notation of Stratton
(1941). Reasoning like that for the cylinder gives, for the curling
mode of smallest X, n = 1 and к = x\/b, where b is the sphere
radius and is the smallest root of ji'(x) = 0, approximately
2.08. Then X = g + Cx^/b2. For the sphere Na = Nb =
Hence for magnetization curling in a sphere
4 Г Cxi2 4
M.H0 = -X + -tM2 = - p + -2- --*M2 , (5-42)
3 b 3 J
Figure 5.2. Reduced nucleation field, H$/2irMS) as a function of reduced
cylinder radius, ЬЛ/в(2/С)^ [5 = radius], for three modes of deviation from
saturation. [After Aharoni and Shtrikman (1958).]
whereas for rotation in unison
M8H0 = -g.
(5-43)
(If we use H rather than Ho, the term in M82 will be transferred
from the first to the second equation, as for the cylinder.) The criti-
cal radius is
at b < bCf rotation in unison occurs; at & > bc, curling. The rest
of the eigenvalue spectrum has been explored (Aharoni, 1959);
there are no modes with smaller eigenvalues. The nonoccurrence of
buckling in a sphere is not surprising: there is no room for a sinus-
oidal variation of amplitude slow enough to avoid excessive ex-
change energy.
Curling in a prolate spheroid would be expected to require a
nucleation field intermediate between those for the cylinder and
for the sphere, and this is found to be so (Aharoni, 1959). The case
of the cylinder with the field not along the axis has also been investi-
gated (Shtrikman and Treves, 1959).
These calculations predict under what conditions a fine particle
should remain single-domain throughout a field cycle, and under
what conditions it departs from the single-domain state. Numerical
calculations give critical radii of order 102 A for iron and nickel cyl-
inders and spheres, in order-of-magnitude agreement with previous
estimates of critical radii for the change from a single-domain state
to a domain structure at zero applied field. Exact agreement of the
two critical radii must not be expected, for two reasons: they are
answers to different questions, and the older estimates were not
rigorous (cf. Sect. 2.6).
Calculations for plane films or plates, infinite in two dimensions,
can be made without difficulty in principle, though in some cases
the search for eigenvalues requires a tedious numerical procedure.
Such calculations should apply to fine platelike particles, as an
approximation, in the same way that the cylinder calculation ap-
plies to fine rodlike particles. The films and plates of the usual ex-
periments, however, have megascopic dimensions in the plane of
the faces and are therefore subject to the perturbing effect of im-
perfections, so that the theory of an ideal film in most cases is not
applicable to them. An exception is the case of a film with the field
parallel to the faces and a direction of easy magnetization normal
to them; then the nucleation field is positive and large. Figure 5.3
shows calculated nucleation fields for such a film (Brown, 1961b),
together with a few experimental points (Huber and Smith, 1959).
The data for large positive nucleation fields fit the theory if the
Figure 5.3. Reduced nucleation field, Яо/4я-ЛГв, as a function of reduced film
thickness, 2bMe(ir/C)^ [2b = thickness], with reduced anisotropy constant,
д/4тМв2, as parameter. Film normal along Ог/; 0ц = 012 = 0, 022 s — 0 < 0.
О, theory; X, experiment. [After Brown (1961b).]
parameter д/4тгМ82 is approximately 0.05. The anisotropy in this
case is attributed to stresses, and the relevant constant is estimated
from elastic data at 104 to 105 erg/cm3; the estimate д/4тМв2 =
0.05 gives 1.7 X 105 erg/cm3. For plates, where the exchange
energy is less important (though still significant), a quick approxi-
mate method of calculating nucleation fields can be used (Muller,
1961); few data are so far available for comparison.
The film or plate calculation predicts not only the nucleation
field but the wavelength and orientation of the incipient domain
structure. These are in order-of-magnitude agreement with the
fully developed domain dimensions observed in zero field. Direct
observation of conditions at nucleation would provide a better test
of the theory.
Tests of the theory for fine particles are subject to the following
difficulties: most experimental data have been taken on powders
composed of particles with a distribution of shapes, sizes, and orien-
tations, about which there is only incomplete information; more-
over the magnetic interaction of the particles has so far defied
rigorous treatment. Nucleation-field theory has contributed to the
understanding of such powders by showing clearly that under cer-
tain conditions an instability of particle magnetization can occur
at a smaller reversed field than that necessary for rotation in uni-
son. But for a full understanding of this phenomenon, we must con-
sider the stages of the magnetization reversal process subsequent
to nucleation; and this requires investigation of the nonlinear range.
The further discussion of fine particles will therefore be deferred to
Section 6.4.
5.5 Approximate Nucleation-Field Calculations
There are cases in which the nucleation-field problem leads to a
very complicated set of partial differential equations, so that a
rigorous solution is very difficult. One such case is the anisotropic
film discussed in Section 5.4; another is an infinitely long cylinder
or prism whose cross section is of noncircular shape (for example,
square). In such cases, approximate methods are useful.
It is often possible to do two approximate calculations of which
one errs, if at all, upward and the other, if at all, downward. When
— M8H is being minimized, imposition of any additional constraint
(e.g., that there must be no poles) produces a result of the first type;
omission of any nonnegative term (e.g., the magnetostatic self-
energy) produces one of the second type. Thus the value of —М&Н
can be placed between definite upper and lower bounds.
The examples mentioned—suppression of poles and simple neg-
lect of their energy—simplify the calculation by eliminating the
auxiliary potential problem altogether. If such drastic methods give
upper and lower bounds that are too far apart, approximate meth-
ods of solving the potential problem can be used (Brown, 1962a).
For given M(x, y, z), the functional
WH = U’dr - JH' (5-45)
if maximized under the constraint that H is irrotational, makes H
equal to the actual magnetizing force H' due to M; Wh itself then
reduces to
*f m
(5-46)
which is the usual “magnetostatic self -energy.” The functional
WB = +
87г
B-Mdr,
(5-47)
if minimized under the constraint that В is solenoidal, makes В
equal to the actual flux density В' = H' + 4тгМ due to M; Wb it-
self then reduces to
(5-48)
which differs by a known constant quantity from Wm. Maximiza-
tion and minimization under additional constraints, as by assuming
a form for the scalar or vector potential with adjustable parameters,
therefore give a lower bound to Wm and an upper bound to W'm,
and hence upper and lower bounds to Wm. These values may then
be used instead of the exact ones in the minimization of
These methods have been used for setting preliminary bounds to
the nucleation field of the anisotropic film (Brown, 1961b) and for
obtaining upper and lower bounds to the nucleation field of an
idealized “whisker” of square cross section (Brown, 1962b).
Chapter 6
Static Problems: Nonlinear
Calculations
6.1 The One-Dimensional Case
When the deviations from uniform magnetization are too large
to permit linearization of the equations, the problem of finding
equilibrium states and testing them for stability is extremely diffi-
cult. Only in a few special cases are analytic methods applicable.
One such case, which we shall examine in this section, is that in
which there is variation along only one Cartesian coordinate. The
calculations of Bitter (1936, 1937) and Shirobokov (1939, 1945)
were of this type. The Landau-Lifshitz wall calculation is also
formally one-dimensional, though—as we shall see in Section 6.2—
it has certain essential features that are implicitly three-dimen-
sional.
To examine stability as well as equilibrium conditions, we start
with the free energy G. We take the z axis as the direction in which
variation can occur. Then if there is to be no variation with x or y,
the specimen must be of unlimited extent in these directions, and
its bounding surfaces must be planes z =« const. Its volume is
therefore infinite, and so is its total free energy. As far as the ex-
change and anisotropy energies are concerned, this fact causes no
difficulty: these energies, being due to short-range forces, are as-
sociated additively with megascopic volume elements and are de-
scribed phenomenologically by energy densities. Therefore we can
consider, instead of the whole specimen, a representative part
contained within a cylinder whose generators are parallel to Oz.
Conveniently we take a cylinder of unit cross-sectional area. Then
the exchange and anisotropy forces contribute a free energy per
unit area normal to Oz
where zr < z < z2 is the region occupied by the specimen.
With respect to the magnetic terms in (?, care is necessary.
Dipole-dipole forces are not short-range. Consider a uniformly
magnetized ellipsoid with principal semiaxes a, b, c, oriented along
the Cartesian coordinate axes, in a uniform applied field Ho- Inside
it the magnetizing force H has components Hqx — NXMX, HOv —
NyMyi HOz — N2M2, where the 2V’s depend on the ratios a:b:c. If
we keep the shape fixed and let the size of the specimen become in-
finite, the values of HX) Hy, and Hz remain constant, so that their
limiting values for the specimen of infinite size depend on the shape
the specimen had before it became infinite. In mathematical lan-
guage, this means that certain sums and integrals related to infinite
dipole arrays and magnetized bodies are only conditionally con-
vergent. In nonmathematical language, it means that statements
about the dipole-dipole energies of bodies of infinite size, but of
unspecified shape, are nonsense. Such statements are nevertheless
sometimes encountered in the literature.
In the present problem the specimen has been made infinite
along x and у while still finite along z. It is therefore an ellipsoid
with a and b infinite but c finite; and regardless of the ratio b:a
during the process a —> 00 and b —> <x>, Nx = Ny = 0, Ng = 4r.
If we later allow z\ and z2 to become infinite, the results will apply
to a specimen that first became infinite along x and у and later be-
came infinite along 2, and not necessarily to one that became in-
finite by some other process.
With M a function of z, we regard the specimen as made up of
layers dz of infinite extent along x and y, each uniformly magnet-
ized. The mutual energy between a given dipole m, at (z, y, z), and
all the other dipoles in the infinite plate is, in the Ix>rentz approxi-
mation, —m*(H' + frM). If we now insert a factor |, sum over
dipoles within our representative cylinder, replace summation by
integration, and drop the constant term involving M2 (cf. Sect.
3.4), we get an expression for the variable part of the energy of
pairs of dipoles inside the cylinder, plus half the energy of pairs of
which one member is inside and the other outside:
Am= - jf M-H'dz. (6-2)
•IZ1
This is not energy localized in the representative cylinder. It may
nevertheless be treated as such in our one-dimensional equilibrium
and stability calculations; for in all variations to be considered, the
dipoles outside the representative cylinder are constrained to ro-
tate in synchronism with their companions inside and at the same
z, and from this it follows that (6-2) gives the correct torques.
Now the one-dimensional magnetization distribution produces
no poles except poles spread uniformly over infinite surfaces z =
const, with volume density —dMJdz and with surface densities
+MZ at z2 and — Mz at zv Such a distribution produces fields
Hfx = Hry = 0, Hrz = — ±тгМz (or B'z = 0). The magnetostatic
energy (6-2) therefore reduces to
An = -if 5МгН'г<1г = 2r f 2 Mt2dz = 2*M2 f * y2dz (6-3)
•'Zl •'«i •'zi
and is equivalent to a uniaxial anisotropy-energy density like the
first term [Kt(l — y2)} in (3-31), with K\ = —2irMe2. Such anisot-
ropy favors orientation of M perpendicular rather than parallel
to Oz but has no tendency to produce a distribution nonuniform
with respect to 2, i.e., a domain structure. The magnetic forces do,
of course, tend to break up the uniform pole layers in each dz into
distributions nonuniform in x and y; but this tendency has been
suppressed by our constraint of one-dimensionality. The foregoing
argument should cause us to be skeptical about the stability of
magnetization distributions variable in only one dimension.
The energy in the applied field causes no difficulty, if we start
with a finite body and let a and b become infinite. The mutual
energy between the dipoles in our representative cylinder and the
ideal permanent magnet that may be considered the source of the
field is — | H0*Mdz. The total Gibbs free energy associated with
J 21
unit од-агеа may therefore be written (for the present we neglect
surface anisotropy)
G
+ 0'2 + t'2) + f(a>0)}dz,
(6-4)
where a' = da/dz, etc., and where
/(«, 0) = wrt(a, 0) + 2irM2y — M8(HQxa + + HQzy). (6-5)
The function f(a, 0) may be regarded as an effective anisotropy-
energy density that includes the orienting effects of internal and
external fields. There is no need to solve an auxiliary potential
problem.
We may use as variables either a and 0 or the angles Ф and 0,
with у = ±[1 — a2 — /?2]^ — cos 0. The exchange-energy part of
the integrand in (6-4) is
= - C[0'2 + sin2 0 • Ф'2],
2
(6-6)
where primes denote differentiation with respect to z, This term
is a nonnegative homogeneous quadratic function of the z deriva-
tives of the chosen variables; it is positive definite, apart from the
physically meaningless case 9 = О, Ф' 0.
Equation (6-4) is mathematically identical with Hamilton’s
integral for the motion of a particle of mass C on the surface of a
sphere of unit radius, with potential energy —/(a, /3) (note the
minus sign). In this dynamic analog, z is time; the rectangular co-
ordinates of the particle are (a, /3, 7); its angular coordinates are
(Ф, 0); and a minimum of potential energy corresponds to a maxi-
mum of the effective anisotropy-energy density /. Variation of
Hamilton’s integral for the particle leads only to differential equa-
tions of motion, since the coordinates at “times” Z\ and z2 are speci-
fied. Variation of G in the magnetic problem leads to the analogous
differential equations (the volume-torque equilibrium equations)
and also to the boundary conditions (surface-torque equilibrium
equations)
Ca = C/3' = 0 at z = zi and at z = z2, (6-7)
since in general no constraints are imposed at the boundaries. A
possible motion of the particle in the dynamic analog does not cor-
respond to an equilibrium solution in the magnetic problem unless
the particle is released at time z^ with zero initial velocity, and from
such an initial position that it comes to rest again precisely at time
*2-
It may be that f is a function of z not only implicitly, through a
and /3, but also explicitly, because of a variation of the anisotropy
constants with z. This corresponds in the dynamic analog to a
system with a potential energy explicitly dependent on time, e.g.,
one driven by a time-dependent electric or magnetic field. For the
present we exclude such cases.
Then a state of uniform magnetization along a direction of min-
imum/ is always a state of stable equilibrium, within the constraint
of one-dimensionality. If this direction is perpendicular to Oz, the
state remains stable when the constraint of one-dimensionality is
removed (see Sects. 5.2 and 5.3); but if the direction has a com-
ponent along Oz, removal of the constraint may cause instability
with respect to a deviation nonuniform in x and y, since such a
deviation breaks up the uniform pole sheets and decreases the mag-
netostatic energy. Except in the presence of a large applied field,
there will be at least two directions of minimum/ (since for Ho = 0,
a reversal of M does not change /). One of these minima is the
deepest minimum. Uniform magnetization in this direction gives /
its smallest possible value and makes the exchange term in G zero.
Any state of one-dimensionally nonuniform magnetization is one
of higher G and can therefore be at most metastable. It corresponds
in the dynamic analog to oscillations of the particle about a min-
imum of potential energy; and this means a spatial oscillation of M
about a direction of maximum anisotropy energy, a physically
implausible situation. These considerations raise considerable
doubt about the physical significance of the nonuniform one-di-
mensional solutions in cases where a stable uniform state exists.
It can in fact be shown that under the conditions we have as-
sumed—no surface anisotropy and no spatial dependence of the
anisotropy constants (or of C)—all nonuniform one-dimensional
equilibrium states are unstable. We shall merely outline the proof
(Brown and Shtrikman, 1962). Consider a variation of the form
bai = eYUi + e2Vi (i = 1, 2), (6-8)
where ti and e2 are arbitrary small quantities, and where щ and
are functions of z; we write <q and a2 for a and 0. Such a variation
from an equilibrium state produces a variation of G which, to the
second order in and e2, is of the form
bG = v]eie2 + -422[v]e22}; (6-9)
the A’s are functionals of the indicated functions. For the special
choice щ = a/, it can be shown that = 0 and that, unless
a/' = a2" = 0 at Z\ and z2, functions rt- can be found for which
A12 0. It then follows from the theory of extrema in two inde-
pendent variables that by suitable choice of q and c2, bG can be
made negative. Thus an equilibrium state is unstable unless all the
quantities «/'(zi), »2"(zi), ai"(z2), and «/'(*2) are zero. But if
= 0, the particle in the dynamic analog starts
with zero initial acceleration as well as zero initial velocity ; it is
therefore at rest at a position of zero force (df/dai = df/da2 = 0)
and will remain there, and the corresponding solution in the mag-
netic problem is a uniform solution. Its stability depends simply
on whether the extremum of f is a minimum.
The proof fails in case / depends explicitly on z; this includes, as a
limiting case, the presence of surface anisotropy. In these cases there
is one nonuniform equilibrium state that arises from each stable
uniform equilibrium state when a variable term in G is introduced
as a small perturbation; by continuity, these states will be stable
at least for sufficiently small perturbations. There will also be states
that arise from the unstable equilibrium states. A large enough per-
turbation may stabilize such a state or destabilize an originally
stable state.
Bitter (1936, 1937) recognized that his one-dimensional distribu-
tions would collapse unless maintained by surface torques; he
speculated that such torques might be present under actual condi-
tions. In view of the instability theorem just outlined, it is clear
that whatever significance the one-dimensional nonuniform equi-
librium states may have in perturbed systems, they have none in
ideal, homogeneous, truly one-dimensional systems.
6.2 The Interdomain Wall
The Landau-Lifshitz wall calculation, though formally one-
dimensional, is not of the type just discussed. In it the boundary
conditions do not arise out of the variational procedure but are
arbitrarily specified at the start: M must be in one specified direc-
tion at one boundary (e.g., at z = — qo) and in another direction at
the other (e.g., at z = +<?©). This specification excludes uniform
magnetization as a solution. The justification for the specification
must come from other sources than the one-dimensional calculation
itself.
To justify it, we return to the general three-dimensional situa-
tion and begin by neglecting exchange and anisotropy energy (in-
cluding the applied-field term); we attempt to minimize the in-
ternal magnetostatic free energy by itself. This acquires its smallest
possible value, 0, when there are no poles. We can achieve this in a
variety of ways, provided we allow discontinuous solutions; and
in the absence of exchange energy, discontinuities of the tangential
component of M involve no energies and are therefore permissible.
Thus the arrangements shown in Figure 6.1 make the magneto-
Figure 6.1. Domain structures (with discontinuities of magnetization) for which
the internal magnetostatic energy is zero.
static energy zero. The normal component of M is zero at external
surfaces and continuous across internal surfaces; and if any con-
tinuous spatial variation of M is included, it is a divergenceless
variation.
We now introduce anisotropy and exchange forces. If we are
careful to choose the magnetization directions in Figure 6-1 as
the directions of easy magnetization as far as is possible, the ani-
sotropy forces will not disturb the pattern much. But if we then in-
troduce exchange, the discontinuities are no longer permissible;
they must be replaced by continuous transitions. If the “domains”
in Figure 6-1 are large enough and the exchange effects small
enough, we can hope to replace each discontinuous “wall” by one
whose thickness is small in comparison with the domain dimensions,
and therefore with the lateral dimensions of the wall. In calculating
the properties of such a wall, we can look at a representative section
}f it on a smaller scale and use the methods of one-dimensional
nicromagnetics; but in such a calculation the orientation of M at a
great distance from the wall on either side is specified in advance.
Some such argument as this is necessary to justify the standard
wall calculation. A necessary condition for its validity is that the
result must be consistent with the assumptions: the calculated
thickness of the wall must be small in comparison with its other
dimensions. Such justification and tests are seldom given in the
usual domain-theory calculations; in fact, the wall concept is often
used when the condition mentioned is not satisfied, e.g., in very
thin films. Without either this or some alternative justification,
such calculations must be regarded as mere guesses, and experi-
mental evidence alleged to support them should be scrutinized
critically to be sure that the reasoning is not circular.
The details of the wall calculation have been presented, in the
original and in various modified cases, so often that there is no
point in reproducing them here; but a few sometimes overlooked
points will be mentioned. Internal magnetostatic forces in the wall
are, as in Section 6.1, equivalent to an additional term in the ani-
sotropy energy; and the “applied field” that acts on the wall must
include the field of other walls, in case (as is often true) the model
corresponding to Figure 6-1 does not avoid poles altogether. Cal-
culation of the final magnetostatic energy in such cases requires
care. If 9 is assumed to be fixed and only Ф is variable, there is no
necessity for deriving the Euler differential equation and then find-
ing a first integral of it; this procedure is analogous to finding the
energy integral in the dynamic analog, and the result can be written
down by inspection of equation (6-4):
|C(a'2 + 0'2 + У2) - /(a, 0) = const. (6-10)
This result holds whether both Ф and 0 are allowed to vary, or only
one; but only in the latter case does this single first integral suffice
to solve the problem. If 0 = const = 0o, (6-10) can be written
sin2 0o-Ф'2 — /1(Ф, 0o) = const. (6-11)
Solution for dz/d& = l/Ф' gives an equation that can be integrated
analytically or numerically to give г as a function of Ф. With
boundaries at finite positions Z\ and z2> the result in the simplest
cases can be expressed in elliptic integrals; but with the usual ap-
proximation zi = — oo and z2 = +«, these reduce to elementary
transcendental functions.
Having found dz/d$ in terms of Ф, one can substitute dz =
(dz/d$)d$ in equation (6-4) and integrate it to get the surface
energy-density of the wall, yw.
In the simplest case 9o = *72 and fi == К vqs2 Ф, with Ф =
—тг/2 at —00 and +t/2 at +«. Equation (6-11) gives
i
-Cl — I — К cos Ф = const
2 \dz/
= 0,
(6-12)
since at z — d$/dz = 0 and cos Ф = 0. Integration gives, if
we take z = 0 where Ф = 0,
sin Ф = tanh
and
(6-13)
= G = 2(2CK)*.
(6-14)
The “thickness” 3 may be defined as the distance between the
two points at which the argument of the hyperbolic tangent has
magnitude unity:
(6-15)
Hence
yw/d = 2A, ywb = iC. (6-16)
6.3 Other One-Dimensional Calculations
Because of the instability theorem of Section 6.1, there are only a
few one-dimensional solutions whose physical significance is as-
sured in advance. These are the ones generated from a uniform
magnetization by introduction of a nonuniformity of the properties
of the material.
Several such calculations have been done by Aharoni and his as-
sociates (Aharoni, 1960; Abraham and Aharoni, 1960; Aharoni,
1961). The object was to resolve the coercivity paradox (Sect. 5.2)
by investigation of specific types of “imperfection.” One-dimen-
sional cases were chosen because of the possibility of analytic solu-
tions or, at worst, numerical integration with a single independent
variable. The internal magnetic energy was neglected; for in an
ellipsoid or one of its limiting forms, if sufficient lowering of the
magnitude of the nucleation field cannot be obtained with the term
(8tt)“ 1fH'(1)2dr in P omitted [eq. (5-23)], it certainly cannot be
obtained with this term included. Uniaxial anisotropy was assumed,
with a direction of easy magnetization and the field direction both
perpendicular to the direction of variation; in the notation of Sec-
tion 6.1, Ф is variable but 9 is assumed to have the constant value
tf/2, and
f = K(z) cos2 Ф — HqMe sin Ф.
(6-17)
The boundaries are = 0 and z2 = °°; by a simple argument, the
results can be applied to a specimen with boundaries at The
initial state is Ф = тг/2, with Hq large.
It is necessary first to find the nucleation field and then, if it is
encouragingly small, to examine the behavior after nucleation. This
behavior may be of either of two types: a discontinuous jump to
Ф = — тг/2 at nucleation, or a continuous process followed by a
later jump to Ф = —тг/2. When the “imperfection” assumed is of
finite extent, the value of Ф at z = <» remains тг/2 until the jump
and then suddenly becomes — тг/2; the mean magnetization (in the
field direction) of the semi-infinite specimen jumps from +Ma to
—Ma; and the coercivity is the value of Hq at which the jump oc-
curs.
The following functions K(z) were used: (1) K(z) =0 for 0 <
z < T, = К for z> T] (2) K(z) = 0 for $ < z < Тъ = К for
z > T2, with a linear transition from 0 to К in the intermediate
region Tx < z < T2; (3) a periodic sequence of regions with К = 0
(the last sentence of the preceding paragraph does not apply to this
case); (4) two such regions separated by a normal region. By the
argument of Section 5.2, the nucleation field Hn cannot be positive.
In case (1) ] Hn| = —Hn is close to 2K/Ma for T « Tc = (C/2K)*;
it decreases rapidly for T > Tc. However, before | Hn | decreases
by an order of magnitude, nucleation ceases to produce a discon-
tinuous jump; and the nonlinear calculation shows that the mag-
nitude of the field at which the jump occurs is never less than
0.25(2A/M8). In case (2) the factor 0.25 is replaced by one that
becomes as small as 0.089 by suitable choice of T\ and T2. This
is still not enough reduction to account for observed coercivities.
In cases (3) and (4) only Hn was calculated; it was reduced less than
in case (1).
Evidently this type of imperfection cannot resolve the paradox,
a fact not surprising in view of the specialized character of the im-
perfection. Apart from the one-dimensionality, a limitation that
may be important is that here only the anisotropy constant is varia-
ble, and not the direction of minimum anisotropy energy.
6.4 The Infinite Cylinder
The nucleation field of the infinite cylinder, with 0ц = g22 = 9
and £12 = 0, was calculated in Section 5.4 for two modes: rotation
in unison and magnetization curling. To calculate the whole mag-
netization curve, it is necessary to solve the nonlinear equilibrium
equations for fields algebraically smaller than the nucleation field.
The nonlinear calculation can be greatly simplified by assuming a
mode of magnetization that constitutes a natural continuation into
the nonlinear range of the mode of small deviation determined in
the nucleation calculation. Thus when instability occurs first with
respect to rotation in unison, one assumes uniform magnetization
in the nonlinear range; when it occurs first with respect to curling,
one assumes a magnetization of the form Mp = 0, Мф 0, дМф/дг
= дМф/дф = 0. An equilibrium magnetization curve can then be
calculated. It must be tested for stability. Portions with dMu/dH
< 0 may be rejected as unstable. Other portions can be tested,
by several methods, for stability with respect to deviations within
the mode assumed: for example, in the case of curling, deviations
with bMp = 0, д8Мф/дг = дЬМф/дф = 0. Testing with respect to
deviations from the mode assumed, for example deviations dMp 0
from a curling solution, is much more difficult and has not been
accomplished.
When the mechanism is rotation in unison, the nonlinear calcula-
tion reduces to the Stoner-Wohlfarth (1948) theory of single-
domain particles. For a cylinder with applied field along the axis, an
irreversible 180° rotation occurs at the nucleation field, and the
hysteresis loop is a rectangle. We have seen that rotation in unison
is never quite the exact mode in a cylinder; but since its nucleation
field almost coincides with that for the actual buckling mechanism
in cylinders with b < bc, it is usually adopted as a sufficient ap-
proximation. For a sphere it is exact (Aharoni, 1959).
When the mechanism is curling (b > bc), the nonlinear calcula-
tion consists in solving a differential equation for the angle c de-
fined by = cog = gin e (6-18)
With uniaxial anisotropy (of energy density sin 2t) in an in-
finite cylinder, the differential equation is
1 d / dA (К 1 \ . M8H0 .
----( P — 1 — (----F — I cos t sin c-------sin e = 0.
pdp\ dp/ \C p2/ C
(6-19)
The boundary condition at p = b is dt/dp = 0. At p = 0, since ф
is indeterminate there, we must have Мф = 0, i.e., t = 0. Solutions
must be found by numerical methods.
For К > 0 (Brown, 1958; Aharoni and Shtrikman, 1958) the
only stable equilibrium state found, for H < Hn, is e = тг for
p 0.* Thus an infinite cylinder with К > 0 and with b > bCJ in a
* The fact that e remains equal to 0 at the point p == 0 is a result of the con-
straint to curling; actually, at some stage of the reversal process the central
line of spins would become unstable and rotate to e = r. It might seem that
the infinite de/dp at p = 0 would entail infinite exchange energy; but if e =
irp/h for p < h, — тг for p > hf the exchange energy per unit length is
sin2 el _rt л Г/ A2 . 1 . 9
—=- dS = ttCI (-) 4-—81П2 рдр
p* J Л LXft/ pz л J
which remains constant as h —> 0.
(6-20)
longitudinal field, is a single-domain particle except during the
jump from magnetization along the axis in one sense to magnetiza-
tion along it in the opposite sense; the hysteresis loop is rectangular;
the effect of the curling process is merely to decrease the coercivity.
The possibility of a reversal by curling, at a lower reversed field
than is necessary for rotation in unison, is an important factor in
the behavior of fine particles. It has to be taken into account in the
theory of powders and of fine-particle permanent magnets (Wohl-
farth, 1959).
The result that only uniform longitudinal magnetization of a
cylinder is stable, when the field and the anisotropy both favor such
a state, is analogous to the result that only uniform magnetization
along a direction of minimum effective anisotropy energy is stable
under the constraint of one-dimensionality (cf. Sect. 6.1). It can be
reinforced by similar physical arguments. When the anisotropy con-
stant of the cylinder is negative, there may be a stable nonuniform
magnetic structure; for the anisotropy favors magnetization trans-
verse to the axis, and the magnetostatic energy is less if the trans-
verse magnetization is nonuniform. This situation is similar to that
in the plate with an easy direction normal to the faces, when the
constraint of one-dimensionality is removed. Muller and Wehlau
(1961) have calculated magnetization curves of a cylinder with
К < 0 and found possibly stable nonuniform states. Muller (1961)
has also investigated the initial nonlinear stages in a plate of the
type described, immediately after nucleation, and has found indica-
tions of a stable domain structure.
6.5 Approximate Methods
Numerical integration by a finite-difference method is always
only approximate. It is often desirable to use a less laborious ap-
proximate method, at least for preliminary calculations.
The standard method is the Rayleigh-Ritz method: a/orm of the
solution is assumed, with one or more undetermined parameters,
and G is then minimized with respect to them. This method was
used by Kondorskii (1952) in nucleation-field calculations that
antedated the rigorous ones and that agreed quite well with them.
It was used in preliminary calculations of magnetization curling in
a cylinder (Brown, 1957a); the calculation gave a jump at about
the right field, but to a magnetization considerably short of the
correct value — M8. This shows that the constraints of the Ritz
inethod can cause considerable error.
The method described in Section 5.5 can be applied to nonlinear
as well as linear problems. By these methods, it may be possible
to find two approximate solutions of a nonlinear equilibrium prob-
lem, such that one gives an upper bound and the other a lower
bound to the correct minimized value of G. But this in itself does
not enable one to set upper and lower bounds to the values of
У, z) at various points. If the two approximate solutions, as
well as the two values of G, are nearly identical, one expects by
continuity that the correct solution will also be nearly identical
with the two approximate solutions; but there is no obvious way
to make the argument quantitative.
All of standard domain theory consists of approximate minimiza-
tion of G, The procedure is usually a Ritz minimization based on a
particular model with adjustable parameters. The chief difficulty
is often the evaluation of the magnetostatic energy. Since the
model itself is highly oversimplified, solution of the potential prob-
lem by elaborate techniques (e.g., Fourier analysis) is hardly
justified; but it is also undesirable to resort to extremely crude
methods (e.g., replacement of parallelopipeds by ellipsoids). In
this application, the methods of Section 5.5, which set upper and
lower bounds to the magnetostatic energy, may be useful. It should
also be noted that if one expresses the magnetostatic energy as
Wfm rather than Wm and then replaces W'm by Wb, one can per-
form a simultaneous approximate minimization with respect to В
and to M (within the constraints of the model, including of course
the constraint M2 = M82). This is a more consistent procedure
than to minimize Wb exactly (by rigorous solution of a potential
problem), yet minimize the total free energy G only to a crude
approximation (by variation of a few parameters).
Figure 6.2. Approximate theoretical magnetization curves for a finite
cylinder, of length-to-diameter ratio 1, with magnetization constrained to re-
main in planes through the cylinder axis: calculated by a Ritz method with 12
independent parameters; Ьд — Hq/M8, boo = h/M8. The curve labeled
“anisotropic” is calculated for uniaxial anisotropy with K\ — 4 X 106 erg/cm3,
M8 = 1500 emu (cobalt at room temperature). [After Halverson (1961).]
In Figure 6.2 we give theoretical magnetization curves for a finite
cylinder, of length equal to its diameter, calculated by the Ritz
method. This calculation (Halverson, 1961) was made primarily
to explore the possibilities and is subject to the following limita-
tions: the exchange energy was neglected except for the require-
ment that M be continuous; the magnetization was constrained to
have only longitudinal and radial components; and the series were
truncated after a small number of terms to permit hand calculation.
Consequently there are many severe constraints, and the calculated
coercivity is larger than for rotation in unison. But the method in
principle is capable of indefinite improvement by increase of the
number of terms in the series, since the functions used form com-
plete sets.
Chapter 7
Dynamic Problems
7.1 The General Problem
This chapter deals mainly with the steady-state response to
alternating applied fields and with the modes of natural oscillation.
Such problems yield to analytic treatment only when the govern-
ing equations are linear or, at worst, subject to small nonlinear
perturbing terms. We therefore assume that the linearized equa-
tions of Section 4.6 apply.
Satisfaction of this condition normally requires the application
of a large constant applied field H0(0) = H0(0)k to insure that the
deviations from a uniform magnetization M(0) = M8k are small.
In certain cases, anisotropy forces can replace the applied field as
a means of satisfying the condition. However, the anisotropy-
energy density as a function of the direction of M always has at
least two minima, whereas the external magnetic energy density
— M • Ho has only one. Ordinarily, if Ho = 0, the internal magnetic
forces, with their abhorrence of poles, create a domain structure,
with different regions magnetized close to different anisotropy
minima. Since this state has so far been studied only by approxi-
mate methods, we are not yet able to develop a rigorous theory of
oscillations about it. In sufficiently fine particles, however, a single-
domain structure is maintained by the exchange forces. Such a
particle in zero applied constant field is in essentially the same
situation, with respect to small alternating applied fields, as is an
ordinary specimen in a large applied constant field.
Resonance calculations have, of course, been made for specimens
in a multidomain state. Such calculations involve the approxima-
tions and uncertainties of domain theory and are therefore not
within the scope of this book.
Under the conditions assumed, the behavior is described by
equations (4-36)-(4-39). By virtue of the linearity, the effects of
static irregularities of the K9 terms on the right of (4-38) and
(4-39), and of the alternating field h0 superpose; we may therefore,
in computing the effect of h0, set and the right members of
equations (4-38) and (4-39) equal to zero. Since the surface anisot-
ropy is small, we shall also set Ke = 0 in the left members of equa-
tions (4-38) and (4-39); however, these terms may be important in
some cases (Kittel, 1958; Seavey and Tannenwald, 1958).
The applied constant field, an anisotropy force, or a combination
of them plays the role of a restoring force with respect to the
generalized coordinates a and fl. In most cases the field dominates.
In the single-domain particle, however, the anisotropy term may be
the dominant one or even the only one.
We assume an alternating applied field h0 = hole10** and seek
steady-state solutions u » Uiewf or a = fl — flie™*.
7.2 Special Cases
All of the following cases have been treated extensively in the
literature, but usually without reference to the closely related
equilibrium problems.
a. The Uniform Mode
We assume an oscillation by rotation in unison, «i = const and
fli = const. Then a and fl automatically satisfy the boundary condi-
tions (4-38)-(4-39) (if Ка = 0). The first-order pole-field intensity
will be uniform if the specimen is an ellipsoid; we assume
that this is so and that a principal axis of the ellipsoid lies along Oz.
To maintain the uniformity, the alternating applied field must
also be uniform. If the x and у axes coincide with the other principal
axes of the ellipsoid,
H\(1) = -MM + N^j) = -ад.И1, (7-1)
where, in the axis system described,
0 '
Nv.
(7-2)
Equations (4-36) and (4-37) therefore give
fc + 0ii + MS2NX + uaM2ii\ax + 0i2 +
------ Pi
Vo J
i&M 8
012-------
Vo J
— 2l/e^oix,
(7-3)
«1 “h [к “h 022 + № у iwMs n]^i — Msh’Qlw
(7-4)
In the limiting case of vanishingly small dissipation, the condi-
tion for resonance is the same as the condition for steady oscillation
with no driving field. The resonance condition can then be found
by setting fcois = ^oii/ = 0 and у ~ 0. For compatibility of the
resulting equations,
= 0 (7-5)
or
(MeH2(0) + 0ii + Ms2Nx)(M8H^ + 022 + Ms2Ny) - 0122
This determines the resonance frequency о)/2тг for given Hz(0) or
the resonance field Hz(0) for given frequency. The resonance value
of the applied field is = HZ(Q) + NZM9. Equation (7-6) in-
cludes various cases originally treated by Kittel (1947, 1948) and
Snoek (1948).
By assuming that the resonance condition is satisfied and solving
for ft • ai, we may find the locus traversed in the (a, /3) plane in a
natural small oscillation. If the principal axes of the anisotropy
tensor (дц) coincide with those of the ellipsoid, so that gl2 = 0, the
ratio :«i is pure imaginary; that is, there is a phase difference of
ir/2 between a and /3, and the locus is therefore an ellipse with
these same principal axes. The ellipse degenerates to a circle when
<7ii + M2NX = $22 + M2Ny. When 012 # 0, the principal axes of
the ellipse are rotated with respect to the ellipsoid axes.
The response at and near resonance can be found by solving the
inhomogeneous equations (7-3) and (7-4) for cq and 3i. The result
is of the form
М8иг = x'-hoi, (7-7)
where the apparent-susceptibility tensor xz is not symmetric.
When ij = 0 and gi2 = 0, it is of the form
x'n — it'
LtK X 22 J
(7-8)
with х'п> X22) and к' real. This type of response to alternating fields
has been investigated in detail in connection with cavity-resonator
methods for studying ferromagnetic resonance, and in connection
with applications of ferrites to nonreciprocal circuit devices
(Soohoo, 1960).
The behavior of a powder of single-domain particles in the de-
magnetized or the remanant state can be studied by setting =
0, or = —NgM8. When “shape anisotropy” dominates, i.e.,
when the gt/s are small in comparison with M2NX and M2Ny,
equations (7-3) and (7-4) become for an ellipsoid of revolution
about Oz
о <*M8
M8\Nx - Ng) +-----
L Vo
+ uaM2fq
[«1 + t£i]
M8[hoix “b (7-9)
from which the elements of the apparent-susceptibility tensor of a
particle are easily found. Usually the particle axes are randomly
oriented with respect to the applied alternating field, and the com-
ponent of alternating magnetization measured is that in the direc-
tion of the applied alternating field. This requires insertion of ap-
propriate angular factors and an averaging over orientations. Such
averaging neglects magnetic interactions between the particles; it
is valid if the powder is sufficiently dilute. The resulting suscepti-
bility tensor for the powder as a whole is, when rj = 0, of the gyro-
tropic form (7-8) (with x'22 = x'n) in the remanent state but re-
duces to a scalar susceptibility in the demagnetized state. Since
the particles are never all of the same shape, a further averaging
over shapes is necessary; this involves a distribution function ф(и)
such that the fraction of the total particle volume that is occupied
by particles with Nx — Nz in the interval и to и + du is ф(и)йи.
For rj 0, the imaginary part of the susceptibility, plotted as a
function of frequency, gives directly, by a simple change of the
scales, the plot of ф(и) against u. This method has been used to
compare particle shapes of different powders (Brown, Hanton, and
Morrish, 1960).
Equations (7-3) and (7-4) with hOi = 0 but with rj > 0 may be
used to investigate the stability of the reference state a = 0 = 0
with respect to rotation in unison. Equation (7-6) is now replaced
by an equation in which each of the quantities in parentheses con-
tains an extra term The values of a? thus determined are
complex; when g22 = {7in 012 = 0, and Ny = Nx, they are
with
M8
To
+ iM82ij >
C11 = {Z11 + М8я/О) + M2NX
(7-Ю)
= Й1 + + (Nx - NZ)M,].
(7-П)
The reference state is stable or unstable according to whether the
imaginary part w" of w is positive or negative, since e**'=
e**4^"*. The sign of w" is the sign of Сц. The stability condition
is therefore Сц > 0. The condition for breakdown of stability,
сц =0, is equivalent to equation (5-32). This approach to the
equilibrium problem gives more information than the static
method, namely information about the initial behavior when the
equilibrium is made unstable by a sudden change of H$. More
will be said about this in Section 7.6.
b. Magnetostatic Modes
If exchange forces are neglected, C in equations (4-36)-(4-39)
may be set equal to zero. Then (if K8 = 0) there is no longer a
boundary condition to be imposed on u. Elimination of «i and
shows that Ui satisfies an anisotropic linear homogeneous second-
order partial differential equation in r, Laplace’s equation outside
t, and linear homogeneous boundary conditions on S. The solution
of this boundary problem determines the natural oscillations in the
limit C = 0. The approximation is good as long as the specimen
dimensions are so large, and the number of nodes in the specimen
so small, that the spatial variations of ui produce a negligible ex-
change contribution in equations (4-36)-(4-39).
These “magnetostatic” modes were investigated theoretically
by Walker (1957). They have since been the object of considerable
theoretical and experimental research (White, 1960).
c. Periodic Boundary Conditions; Spin Waves (Soohoo, 1960)
At the other extreme, when the spatial rate of variation is large,
the conditions imposed on the alternating part of the magnetization
at the surface of the specimen will have little effect except within a
distance from the surface comparable with the distance between
nodes. In this limiting case, conditions in the interior can be studied
by replacing the actual boundary surface and the correct boundary
conditions by more convenient ones.* Especially convenient are
the periodic boundary conditions /(r + iLx) = f(r + jLy) —
У(г + kLz) applied to f = a, ft, and U. A solution can then be
* It must be emphasized that this statement applies only to the alternating
parts H<x)) of M and of H. In the calculation of the constant components
of M and of H (i.e., of their values M(o), H(o) in the reference state about
which the small oscillations M(1), occur), the geometry of the specimen
must still be taken into account.
found by assuming a spatial variation of the form e *Kr, with
KXLX ~ N\2tt, KyLy = Л^22тг, KZLZ = W32t (2Vj, Л^2, Я3 inte-
gers). The compatibility conditions for the a, 0, and U equations
then determine the possible values of w for given K.
This plane-wave approximation has been used as the basis of
much theoretical work. Since cylindrical, spherical, and other waves
can be built up by superposition of plane waves, the plane-wave
approximation actually has considerable generality within the
range in which the basic boundary approximation is justified; this
is the range | К | L 2т, where L is a representative linear dimen-
sion of the specimen. The approximation fails in the range | К | L ~
2т. In the usual “spin wave” theory, the uniform mode, discussed
in Section 7.2a, is treated rigorously and assigned the wave vector
К = 0; all other modes are treated by the plane-wave approxima-
tion. This procedure has the great merit that it permits analytical
treatment of problems otherwise intractable. One should remem-
ber, however, that it involves certain approximations.
7.3 Resonance Fields and Nucleation Fields
We may regard the boundary-value problem of Section 7.1,
with Я0(0) = in general ^0, as determining either the res-
onance frequencies for given applied fields or the resonance fields
for given frequency. The latter point of view corresponds to the
usual experimental situation, in which the frequency is fixed by
the geometry of a resonant cavity and the field is adjusted for res-
onance by variation of an electromagnet current.
If this latter point of view is adopted, the dynamic problem of
determining the resonance fields and the corresponding modes of
oscillation is closely related to the static problem of determining the
nucleation fields and the corresponding modes of initial deviation
from uniform magnetization along Oz, In fact, by comparison with
the equations of Section 5.3 it is easily seen that the nucleation fields
are the zero-frequency limits of the resonance fields, and that the
modes of initial deviation are the zero-frequency limiting forms of
the modes of natural oscillation. This relation has already been il-
lustrated in detail, in Section 7.2a, for the uniform mode (rotation
in unison).*
A formal calculation of undamped natural frequencies and modes
will yield some that correspond to a precession about a minimum
of potential energy and some that correspond to a precession about
a maximum. If a small damping term is introduced, the precession
will decay in the former case and will increase in amplitude (and
ultimately pass beyond the linear range) in the latter. This method
could be used, instead of that developed in Section 5.3, as a means
of testing the stability of the initial uniform magnetization Mak.
All formally calculated resonances that correspond to values of
Я0(0) algebraically smaller than the largest nucleation field may be
rejected as physically without significance; they cannot occur be-
cause the reference state about which the oscillation occurs is un-
stable with respect to other modes of deviation, if not with respect
to the one under consideration.
t
7.4 General Theorems
The following theorems relate to the natural modes of undamped
oscillation. We set hoi and q in (4-36) and (4-37) equal to zero and
rewrite the equations
CV2u — ku — Г*и + iQ X u + h = 0, (7-12)
* If one attempts to measure the resonance field (for, say, the uniform mode)
as a function of frequency all the way down to zero frequency, one encounters
the following difficulty: because of imperfections, nucleation occurs first with
respect to some very complicated non uniform mode (i.e., a domain structure
is nucleated), before the nucleation field for the mode under observation is
reached. This difficulty would not occur with a specimen consisting of a single
(perhaps carefully selected) single-domain particle. At present the methods of
observing resonance are not sufficiently sensitive to permit observation of an
isolated particle; and in a powder of fine particles, the resonance is spread out
by the distribution of shapes and orientations. Conceivably it may sometime
become possible actually to carry out the contemplated experiment by prepara-
tion of nearly flawless specimens, by observation of single particles, or by
precise control of particle shapes and orientations.
with
M8
K = Q = —wk ok, h = MsH'i(1);
7o
(7-13)
Г is the 2-by-2 tensor with components дц. We have dropped the
subscript 1 in uj to simplify the notation.
We first suppose that the frequency is given and that the res-
onance fields are to be found. Then the eigenfunction u№ satisfies
the differential equation
CV2un — кпип — Г-ип + t’Q X un + hn = 0 (7-14)
and the boundary condition Cdun/dn = 0. It can be shown with-
out difficulty that the eigenvalues кп of к are real and that the eigen-
functions un are orthogonal in the usual sense, i.e.,
J un**uwdr = 0 if кт кп, (7-15)
where un* is the complex conjugate of un.
To prove this, multiply (7-14) by uw*« and integrate over t;
interchange m and n and take the complex conjugate; subtract the
second result from the first; and use Green’s theorems, the sym-
metry of Г, and the magnetostatic reciprocity theorem of Section
3.5 to transform the result to
(7-16)
For m = n, this shows that кп* — кп = 0, i.e., that кп is real; for
m # n, it then gives (7-15). Degenerate sets of eigenfunctions, if
there are such, can be orthogonalized by linear combination in the
usual manner; then (7-15) will hold for any two of the eigenfunc-
tions, whether or not кт И кп. The functions may also be normal-
ized, so that
Uw**Un(/T = 8mn. (7-17)
On the supposition that the set of eigenfunctions un is com-
plete, an arbitrary vector function u, defined inside r, can be ex-
panded as a series
U = 22 cnUn (7-18)
with
n
(7-19)
The eigenvalues used may be those corresponding to an arbitrary
real Q; in particular, to Q = 0. The Kn’s are then (in M8H units)
the nucleation fields, and the un’s are the modes of deviation from
saturation (Sect. 5.3).
We now turn to the case in which the applied field is given and
the resonance frequencies are to be found. This case is interestingly
different from the usual one. The eigenfunction up satisfies the
differential equation
CV2up — kup — r*up + Шрк X Up + hp = 0 (7-20)
and the boundary condition Cdup/dn = 0. The procedure used be-
fore leads to the result
For q = p, this becomes
Since k*Up X up* — apfip* — (3pap* = 2i Im (ap/3p*), it is not
obvious that — Qp*, rather than the integral, must vanish.
However, by multiplying (7-20) by up*-, integrating, and trans-
forming, we find that
(V/V)-(W]^
1
4тгМв2
y*hp**hpdr.
(7-23)
If к is large enough to make the quadratic form corresponding to
the tensor к + Г positive definite, the right member is real and
positive; then the left member, which is 2QP Im lfapflp*dr], is also
real and positive, so that the integral is nonzero and Qp is real and
nonzero. As к is decreased beyond this point, Qp will continue to be
real and nonzero until the right member of (7-23) becomes zero.
The largest value of к at which this can happen can be found by
setting the right member of (7-23) equal to zero, solving for к, and
maximizing the result with respect to the function up. But this is
precisely the procedure by which the algebraically largest nuclea-
tion field was found in Section 5.3 (it is easily verified that allowing
complex values for up makes no difference in the nucleation-field
calculation). Therefore as long as k/M6 exceeds the largest nuclea-
tion field, every Qp is real and nonzero.
In this case, therefore, the eigenvalues Qp are real provided that
exceeds the largest nucleation field of the static problem. If this
condition is satisfied, it follows without difficulty that the eigen-
functions up are “orthogonal” in the somewhat unusual sense that
We may without loss of generality use the notation up for the eigen-
function corresponding to a positive eigenvalue 12p; then up* will
correspond to — Qp. We can then normalize the up;s so that
(7-25)
on the supposition of completeness, an arbitrary complex function
u can be expanded as
u = ^2 cpup (7-26)
with
p
cp = гк* J*up* X udr.
(7-27)
The character of the eigenvalues when Я2(0) does not exceed
the largest nucleation field is of no physical interest; for then there
is no stable reference state about which the assumed small-ampli
tude precession can occur.
7.5 Example: the Infinite Cylinder
There is one case in which the complete ferromagnetic spectrum
can be investigated without excessive difficulty; this is the circular
cylinder of infinite length and of finite radius b, with a hard or easy
direction of uniaxial or cubic crystalline anisotropy parallel to the
cylinder axis: gn — 922 = g, gi2 == 0. The resonance fields and
modes are the dynamic generalizations of the nucleation fields and
modes studied by Aharoni and Shtrikman (1958).
The simplest modes are the dynamic generalizations of rotation
in unison and of magnetization curling. Rotation in unison was
treated in Section 7.2a; for the cylinder, Nx = Ny = Nb = 2r,
Nz — Na = 0. In magnetization curling, the potential U is most
concisely described by the statement that the flux density vanishes.
The components of u in cylindrical coordinates are (we omit the
factor ewi)
2
Щ = E BJiM, (7-28)
8 = 1
2
UP ~
8 = 1
Cp82 + c
BgJ l(PsP)>
(7-29)
where px2 and p22 are the roots of
(Ср2 + с)2 + 4тгМ/(Ср2 + с) - Й2 = 0. (7-30)
Here
с - К + g = M8H™ + g. (7-31)
The boundary conditions require that either = 0 and
B2 = 0, or Ji(p2b) — 0 and = 0. For either solution (s = 1 or
2), the nth mode has p8 = b/xfn, where x'n is the nth root of
Ji'(x) = 0. Setting pi or p2 equal to b/x'n gives an equation that
determines the variation of c (and hence of Я/о) s Я) with Й
(and hence with o>), or vice versa, for the corresponding mode.
We thus find for the variation of resonance field intensity with
frequency
HM8 = c — g = -g- Cx'n2/b2 - 2tM82
± 2ir>s2[l + (А/2тгМв2)2]и, (7-32)
with the upper sign for s = 1 and the lower for s = 2 if we take
Pi2 > P22 • We may then disregard the solutions s = 2, for they
give values of H below the static nucleation fields for curling and
for rotation in unison. The solution s = 1, n = 1 gives an H that
decreases with decreasing frequency and that reduces at fl = 0 to
the static nucleation field for curling; this mode of oscillation can
therefore occur at an arbitrarily small frequency if nucleation oc-
curs by curling. For s = l and for any n, the ratio ир/иф is ze, with
fl 1
2irM2 1 + [1 + (Q/2vM2)2]'*'
(7-33)
As ш -> 0, ир/иф —» 0; as w —> oo, ир/иф —> i. Thus as the
frequency increases, a radial component of magnetization, in quad-
rature with the circumferential, develops; and what began as a
circumferential oscillation becomes an elliptic precession and ulti-
mately a circular one.
The variation of resonance frequency with de field intensity H
is given, over the permissible range for H, by
w = Y0
g + ta'n2/b2l
M8
и
M8
H + 4ttM8 +
(7-34)
for the nth mode. The lowest attainable frequency for the nth
mode is found by setting H equal to the static nucleation field. In
case nucleation occurs by curling, this frequency is zero for the
mode n = 1 and is attained at H = — [0 + Cx\2/b2}/M8i the
nucleation field for curling. At very large H, « JqH.
Figure 7.1. Variation with frequency of the resonance fields for rotation in
unison and for magnetization curling, in an infinite cylinder: (a), cylinder
radius b less than bc of Fig. 5.1; (5), cylinder radius b greater than 6C. [After
Brown (1959b).]
Figure 7.1 shows the variation of resonance field with frequency
for rotation in unison and for the first mode of magnetization curl-
ing: (a) refers to a particle so fine that the nucleation field for rota-
tion in unison is algebraically larger than that for magnetization
curling; (b) refers to a particle so large that the nucleation field for
curling is larger than that for rotation in unison.
7.6 Transient Problems
If the damping terms are retained in the equations of motion,
but the applied alternating field is set equal to zero, the resulting
natural frequencies are complex. The imaginary part describes a
decay if it is positive, a build-up of the precession amplitude if it is
negative. In Sections 4.5 and 7.2a we saw that such a calculation
could be used, instead of the static method of Section 5.3, to test
the stability of the reference state about which the precession oc-
curs. It may furthermore be used to gain information about the
mode of initial departure from equilibrium when the equilibrium
ceases to be stable because of a change of applied field.
Suppose that the applied field = Hfi?, initially large enough
to make the state a = 0 — 0 stable, is suddenly decreased to a
value algebraically below the nucleation field for certain modes
1,2, • • •, N but still above the nucleation fields for the other modes.
Calculation of the natural frequencies wn will give Im (wn) > 0 for
n > N but <0 forn < A. If we write wn = ы'п + tw"n, the quanti-
ties l/(—w"n) for the modes n < N are time constants for the
build-up of amplitude of mode n after a random disturbance starts
the system away from equilibrium. The mode with the shortest
time constant will soon dominate, unless the function describing
the spatial variation of the initial disturbance is orthogonal to the
eigenfunction un for the mode in question.
The amplitude of such a precession will soon increase beyond the
range of validity of the linear approximation. At this point it is
necessary to go over to a nonlinear calculation. But that calcula-
tion can be simplified by assuming a form that reduces, for small
amplitudes, to the mode found dominant in the linear calculation.
This method is illustrated, for static calculations, by the theory of
finite curling in a cylinder (Sect. 6.4).
In view of the strong dependence of the nucleation fields of actual
specimens on random imperfections, there is some doubt whether
the suggested procedure would be successful in selecting the right
modes. A further possible objection is that the phenomenological
damping term in the equation of motion, with a single frequency-
independent constant 77, may not be sufficiently accurate. How im-
portant these difficulties are cannot be decided except by trying
the method, and this has not been done. Considerable work has
been done on mechanisms of magnetization reversal other than
rotation in unison (Gyorgy, 1960); but it has been guided chiefly
by intuitive guesses and empirical criteria.
Chapter 8
Magnetostriction
8.1 The Voigt Approximation
The term “magnetostriction” is used in a narrow sense to de-
scribe the change of dimensions of an unstressed ferromagnetic
specimen with magnetization. The term is also used in a wider sense
to describe the whole set of phenomena in which magnetic and
elastic quantities affect each other. Here we are using the term in its
wider sense.
Most treatments of magnetostriction are based on the approxi-
mation that formed the basis of Voigt’s (1928) treatment of electro-
elastic and magnetoelastic interactions. That this is an approxima-
tion is usually not explicitly stated.
In Section 3.2 we derived a formula for the work done in a small
change of magnetization of a rigid ferromagnetic specimen [eq.
(3-13)]:
Jh0-8Mdr.
(8-1)
5Ж
If we express the applied field Ho as H (the total magnetizing force)
minus H' (the part of H due to M) and use the magnetostatic
formula (3-48), we get
5Ж
j'H-SMdr + 8Am,
(8-2)
where Am is given by equation (3-28). Since Am is a perfect dif-
ferential, we can either define the thermodynamic “internal en-
ergy” as U of Section 3.2 and use Ho as a thermodynamic variable,
or define the “internal energy” as U — Am and use H as a thermo-
dynamic variable. The latter procedure was usual in formal classical
theories, but it is useful only when H is subject to direct control;
therefore we have preferred the formulas containing Ho. The choice
of H or Ho as variable is not important at this point. What is im-
portant is that neither the formula for 5W in terms of Ho and 5M
nor the formula for in terms of H' and 5M is valid if the speci-
men undergoes strains during the magnetization process. This is
clear from the derivations.
In the theory of the elasticity of bodies subject to small strains
(Love, 1934; Sokolnikoff, 1946), the work done in a small change of
the elastic displacements u,,
TibUjflS
can also be expressed in the form
(8-3)
(8-4)
T f j $0 ijdr
for quasistatic processes. We use a slight modification of Sokolni-
koff’s notation for stresses and strains: Fi (i = 1, 2, 3) is the ith
Cartesian component of the applied body force per unit volume, Ti
(here we omit Sokolnikoff’s symbolv) is the zth component of the
applied surface force per unit area, тц is the jth component of the
elastic force per unit area across a surface normal to Ox^ and
(8-5)
Here and hereafter, summation over repeated subscripts is under-
stood. These formulas depend on the assumptions that all forces on
the specimen can be expressed either as forces F$r on volume
elements or as forces TadS on surface elements, and that the ratio
of the torque on a volume element to the volume of the element
vanishes as the volume approaches zero. The latter property holds
if the only torques present are the torques г X Fdr due to the body
forces F (whose components are Ft); in elasticity theory this is
assumed to be true, and it follows that the stress tensor тц is sym-
metric, i.e., Гц = тц. In the presence of magnetization, however,
couples of the form M X Hodr are present; they do not have the
property just described, and in their presence the stress tensor is
no longer necessarily symmetric. Furthermore the dipole-dipole
forces exerted by one part of a magnetized specimen on another are
of complicated form, and it is not obvious that they can be reduced
to the forms F^dr and T^dS.
In the Voigt approximation, these complications are ignored.
It is assumed that the magnetic work fif still given by equation
(8-1) or (8-2), even in the presence of strains; that the elastic work
is still given by equation (8-3) or (8-4), with the stress tensor still
symmetric, even in the presence of magnetization; and that the
total work can be found by simply adding the two expressions.
In the integrals in (8-1) and (8-2), dr — dxYdx2dx3 is an element of
undistorted volume; Ho is the applied field intensity at a fixed
point (xi, x2, x3); and M is the magnetization at the point occupied
by the material particle whose position before the strain was
(xb x2, x3). Since this particle is now at (xi + ui, x2 + u2, x3 + u3),
Ho and M are being evaluated at slightly different points.
Magnetomechanical interactions are now taken into account by
supposing that the appropriate thermodynamic potential includes,
besides the terms separately characteristic of a rigid magnetic
body and of a nonmagnetic elastic body, terms that contain both
the magnetic and the elastic variables. These interaction terms are
assumed to consist exclusively of volume integrals of functions of
the magnetization and strain components.
Let us scrutinize this last assumption in relation to foriiiula
(3-27), which evaluates the dipole-dipole energy Um before absorp-
tion of the M-A*M term into the “anisotropy” energy. Both parts
of Umi the part containing H' and the part dependent only on local
conditions, will vary with the strains even if M remains constant:
the former because of displacement of the sources of H', the latter
because of the displacement of dipoles near the one considered.
(Whether the Lorentz sphere remains a sphere or is allowed to de-
form with the body affects the second and third terms separately,
but not their sum.) The use of a volume interaction-energy density
takes account of the latter of these effects but not of the former;
the Am of equation (3-28) is still calculated for the rigid body, i.e.,
by the method already explained for the integrals in (8-1) and (8-2),
In consequence of this approximation, the theory being described
will not predict a shape dependence of magnetostriction.
For convenience we call this the “Voigt approximation/’ but the
assumptions just stated differ in detail from Voigt’s because he
used H rather than Ho as an independent variable. The theory we
are presenting was actually developed by Akulov, Becker, and their
followers. The assumptions here stated explicitly were not so stated
in their theories, a fact that has led to considerable confusion.
We shall adopt these assumptions until the beginning of Section
8.5. We shall, however, not presuppose elastic equilibrium (quasi-
static processes); the transformation from (8-3) to (8-4) can there-
fore not be made. We write the first law of thermodynamics, in
generalization of (3-14),
dU
§-J- bQ)dr +J
(8-6)
hence for natural changes, by (3-15),
6U
+ F ibuj +
Т$и$8.
(8-7)
As before, we define A by (3-17); but now
J*+ FiUi + TS')dr - JTiu.dS.
(8-8)
Then (8-7) becomes
0г HibF i
UibT\dS.
From (8-9) we see that at given T, HOo Pif and T«, G in natural
changes can only decrease. Stable equilibrium therefore requires
that G be a minimum for given T, Hq^ Fi, and 7\.
For reversible changes, the equality sign holds in (8-9); further-
more, by use of the elastic equilibrium equations (which we have
not yet derived) equations (8-8) and ф-9) can be transformed to
U
+ Tijeij + TS'jdr,
(8-10)
0г " — S'dT)dr.
Formulas of this type (with Hi replacing HQi), and their electric
analogs, were used in Voigt's theory of crystals, where G was as-
sumed to be the volume integral of a free-energy density that was a
function only of the local thermodynamic variables T, Hi, and
That assumption is incorrect for a ferromagnetic material capable
of domain structure, and we shall make no use of it.
8.2 Free-Energy Formulas
To carry out the minimization, we must assume dependence of A
on certain internal variables and then, by phenomenological or
microscopic methods, arrive at a formula describing this depend-
ence. In keeping with the nature of the Voigt approximation, we do
this by adding to the A of a rigid ferromagnetic specimen, Sec-
tion 3.4, the A of a nonmagnetic elastic specimen, plus interaction
terms involving both the internal magnetic variables and the in-
ternal elastic variables.
The magnetic part is given by equation (3-44) with the terra
—not yet included. The internal variables are the direc-
tion cosines a, fi, у of the magnetization and their spatial deriva-
tives.
The elastic part is the volume integral of an “elastic energy"
density, which to a sufficient approximation can be expressed as a
homogeneous quadratic function of the strains, with coefficients
that are functions of the temperature. The absence of linear terms
is a consequence of the definition of the unstrained reference state,
which is taken to be the stable equilibrium state at temperature T
in the absence of body and surface forces; if thermal expansion
were to be treated, another definition might be preferred. It is con-
venient to introduce the abbreviated notation (essentially that of
Voigt)
T1 = 711, Г2 s T22) T3 = T33, (8-12)
^4 = r23, T5 T3b TQ == 712, (8-13)
= ^11, e2 = e22, ез = e33> (8-14)
e4 = 2е2з, = 2e3i, = 2^12. (8-15)
(Though we are for convenience introducing the notation for the
stresses rt- or nj at this point, they have not yet occurred in our de-
velopment of the theory.) The elastic Helmholtz free-energy density
is then
G'e = (8- Ifi)
with Cji = cu.
Under arbitrary body forces and surface forces 7\, the Gibbs
function of a nonmagnetic elastic body is
(8-17)
where ae is to be expressed in terms of the derivatives dui/dxj by
returning to the original strain notation and using the definitions
of the ei/s. The equilibrium conditions can be found by setting the
first variation 6G equal to zero. After the variations б(ди</джу) have
been eliminated by integration by parts, the coefficients of but in
r and on S may be equated to zero. The result (see Love, 1934,
Chapter VII) is equivalent to the usual equilibrium equations,
derivable by more elementary methods,
(8-18)
where the r’s are defined, in the one-subscript notation, by
dae
?i = = dj&j,
dd
(8-20)
(In our treatment, this is the first point at which the “stresses” r*
or have actually been introduced in the theory.) By similar
steps, the elastic equations of motion can be derived by Hamilton’s
principle.
By (8-18), uniformity of the stresses requires that body forces
be absent, Fi = 0; (8-19) then determines the surface forces that
must be applied to maintain the specified stresses. In a homo-
geneous body, by (8-20), these conditions require uniformity of the
strains also; uniform strains, by (8-5), correspond to displacements
that are linear functions of x, y, z.
The solution of (8-20) is
— SijTj
(8-21)
with (sty) the matrix inverse to (c,y), so that djCjk = then
Sy» = $ij. In Voigt’s terminology, the ct/s are the “elastic constants”
and the s^’s the “elastic moduli.” For a cubic crystal, with the cube
edges as coordinate axes, the matrix of c’s is
~Сц C12 £12 0 0 0
C12 £11 ^12 000
£12 £12 £ц 0 0 0
0 0 0 С44 0 0
0 0 0 0 С44 о
.0 0 0 0 0 £44_
(8-22)
and
(£ц + 2С12)(Зц + 2$12) = (£11 — £1з)(вц — «12)
= С44844 = 1.
(8-23)
То take account of magnetoelastic interaction, we now add to A
terms that contain both the direction cosines a, 0, 7 and the strains
ei (i = 1, 2, •• •, 6). On the assumption that only a volume integral
is needed and that the lowest-order terms of a series expansion are
sufficient, the cubic symmetry dictates a Helmholtz free-energy
density
dme = fco(^i + e2 + e3) + fci(eia2 + e2fi2 + e3?2)
+ 2k2(e^y + е57« + еба/3). (8-24)
Higher powers of the direction cosines are sometimes included.
Inclusion of terms quadratic in the e/s, with coefficients dependent
on a, /3, 7, would be equivalent to allowing the elastic constants
Сц to vary with the direction of M.
Stable equilibrium states are those that minimize the total G
with respect to the direction cosines a, 0, 7 of the magnetization and
the components u2j u3 of elastic displacement, all of these being
functions of xf уfz; the minimization is to be performed at constant
T, Hm, Fif and Ti and is subject to the constraint a2 + ft2 + y2 =
1. This is a complicated problem, and solutions have been found
only in special cases.
It must be realized that once interaction terms have been in-
troduced, the division of any quantity into “magnetic” and “elas-
tic” terms is dependent on the choice of independent variables.
Thus the separation of a strain into an “elastic” and a “magnetic”
part depends on whether the independent variables are the stress
and field components or the stress and magnetization components.
8.3 Static Effects of Magnetostriction
Within the limitations of the Voigt approximation, it is possible
to obtain rigorous formulas for the strains when the magnetization
is uniform and when the applied forces are such as would cor-
respond, in a nonmagnetic specimen, to a solvable problem. For
specified nonuniform magnetization, the problem of finding the
strains resembles the problem of finding the strains in a nonmag-
netic body under specified applied forces; the mathematical diffi-
culties are the same and are in general formidable. If the solution
of this problem could be found for arbitrary M(r) and substituted
back into G, the further minimization with respect to a, p, у would
involve essentially the same procedures and difficulties as for a rigid
body. The complexities of this general problem are obviously great.
We shall discuss only a few relatively simple special problems.
We consider first the strains corresponding to uniform magneti-
zation of a specimen free from applied forces: Fi = О, 7\- = 0. The
variable part of G then reduces to the volume integral of
Ue 4“ Ume — Ti fy, (8-25)
where, by (8-24), 4
Ti* = -(*0 + fcia2), • • •, T4* = -2Wy, • • •. (8-26)
Since the 7$*,s are uniform, G is minimized by uniform e/s that
minimize ae + ame; namely
(8-27)
or in full
61 = SUT1* + 812(t2* + 73*)
= — Wsn + 2$1г) + ^i[$n«2 + $12(02 + 72)]}, • • •, (8-28)
€4 == $4474* = — 2k2S44&y, • • •. (8-29)
The dilatation is
Ci + e2 + 63 — — (3fco + &1)($п + 2$1г)
(8-30)
and is independent of the direction of magnetization. Since the
reference value of the volume is arbitrary (at least as long as ther-
mal expansion is not of interest), it is convenient to take it as the
volume of the stress-free but fully magnetized specimen; this
amounts to assigning fc0 the value —&i/3. Then (8-28) simplifies to
61 = -fci(sn - Si2)(a2 - i), • • ••
(8-31)
From these equations, the elongation in the direction of M can be
found for various orientations of M, and in particular for orienta-
tion along [100] and [111]; and thus the constants fci and k2 can be
related to the saturation magnetostriction constants Хюо and Хщ:
= — ------------= “ - (сц — С12)Хюо, (8-32)
2 — $12 2
к2 —
2 S44
We shall also need the value of ae + ame that results from the
minimization with respect to the strains (that this is indeed a min-
imum follows from the positive definite character of the elastic
energy density ae):
H- O'me ~ T
= COnst + [fci2($n — $i2) — 2fc22«44]
X (/^V + V V + a2^2)
= const + ^[(сц “ C12)X1OO2 2C44X1112]
X (/32T2 + 72a2 + a2/32). (8-34)
The effect on G is simply to replace the zero-strain anisotropy con-
stant Кi in equation (3-30) by the zero-stress anisotropy constant
TC'i = Ki + -j[(cn ~ C12A1002 — 2C44X1112]. (8-35)
The correction is normally a minor fraction of K^.
Since the general expression for G involves the displacements щ
and their derivatives only in quadratic terms with constant co-
efficients and in linear terms with the quantities 77*, and Ti as
coefficients, the general variational procedure at given M (whether
uniform or nonuniform) leads to linear equations whose solutions
can be found by superposing the solutions corresponding to тД
Fi, and T{ separately. The resulting expression for G, however, will
involve F^ and the direction cosines in a complicated way.
Further minimization with respect to a, 7 could in principle lead
to a complete description of the variation, reversible or irreversible,
of magnetization and strains with applied fields and stresses.
8.4 Effect on Nucleation
The question naturally arises: can magnetostrictive forces re-
solve the coercivity paradox of Section 5.2? We shall show that
within the framework of the Voigt approximation, they cannot.
For a specimen under no applied forces (F{ = 0, Ti = 0), at
arbitrary given the variable part of the Gibbs function is
(8-36)
The variational procedure leads to the equilibrium conditions
dry,
----= 0 m r, Tjilj = 0 on S,
dx;
(8-37)
where ae + ame now replaces ae in equation (8-20), so that
— T1 — CijVj T1 j
(8-38)
• • •
^23 = r32 = T4 == C4jCj ~ T4*, • • «.
(8-39)
Solution for the equilibrium displacements and strains is not neces-
sary for our purposes; the equilibrium conditions are useful chiefly
because they suggest introduction of the quantities n. Solution of
(8-38) and (8-39) fdr the e/s in terms of the rfs and tj*’s gives
(8-40)
and substitution of this in (8-36) gives
G - Go = ~ + T3*)dr.
(8-41)
The integrand is
Sv(rtTy + TfTj* — — t/t,*) = Sij(TiTj — (8-42)
since
(8-43)
But substitution of the values (8-26) of the r^’s, with kQ = — fci/3,
shows that
- Кт,*т/ = (X'l - Ki)(/?V + 72«2 + a2/?2) + const, (8-44)
where K\ is given by (8-35). Therefore
+ - KJ J(02y2
+ y2a2 + a2/?2)dr + const.
(8-45)
When this expression is added to the part Gq of G that is inde-
pendent of the strains, the effect is (1) to replace the anisotropy con-
stant K1 at zero strain by the anisotropy constant K\ at zero
stress; (2) to add an integral whose integrand ^Sipyrj is a positive
definite quadratic form in the quantities r,. The argument in Sec-
tion 5.2 therefore proceeds precisely as before; but the lower limit
to the coercivity now involves K\ rather than K^. This could re-
solve the paradox only if K\ were of lower order of magnitude than
Ki; but actually it is of the same order. Therefore magnetostriction
within the limits of the Voigt approximation cannot resolve the
coercivity paradox.
Clearly, inclusion in the n*’s of higher-order terms in a, /3, у will
require merely a change of higher-order anisotropy constants from
their zero-strain to their zero-stress values. Since the anisotropy
constants directly measured are usually those at zero stress, the
conflict between experimental coercivities and the theoretical
lower limit is even more direct when magnetostriction is taken into
account than when it is neglected.
To find the effect of magnetostriction on the individual nuclea-
tion fields would require evaluation of the equilibrium т/s. In view
of the foregoing argument, no order-of-magnitude effect on nuclea-
tion fields can be expected; but there might be an appreciable shift
of the critical size for curling vs rotation in unison. For rotation in
unison, by (8-27), the т/s are zero, whereas for curling they are not;
therefore the positive term in G will increase the nuclea-
tion field magnitude for curling and thereby shift the critical size
upward.
8.5 Criticism of the Voigt Approximation
It was pointed out in Section 8.1 that the conventional treatment
of magnetostriction rests on approximations that are certainly
wrong. Nevertheless Voigt’s theory of piezoelectricity, based on
analogous approximations, has been successful; it might therefore
seem that the errors in the approximation are negligible. However,
in piezoelectric devices it is possible to control the electric field
intensity E reasonably directly by control of the potential difference
between the electrodes; in ferromagnetic experiments the quantity
directly controlled is the applied field Ho, not the magnetizing force
H within the specimen. This fact is especially important when a
domain structure is present, so that H may vary in a complicated
manner even though Ho is uniform in space. Furthermore piezo-
electric phenomena are linear, whereas an essential feature of ferro-
magnetic processes is their nonlinearity.
Failure of the Voigt approximation in ferromagnetic phenomena
is most clearly evident in the magnetostriction “form Effect,” i.e.,
the dependence of the magnetostrictive change of dimensions on the
shape of the specimen. The form effect will be discussed in Section
8.6.
It is possible to develop a self-consistent theory free from Voigt’s
approximations; but to do so requires a careful analysis of the
meanings of “magnetic” and “elastic” force, a redoing of the theory
of elasticity with magnetization taken into account, and a recogni-
tion that there is considerable arbitrariness in the definition of
stress. The results (Brown, 1951b) are briefly as follows. The rate
of work is
DM
--dm
dt
2tMx2)
d^23
dt
* dr,
(8-46)
where dm is an element of mass, M is magnetic moment per unit
mass, D/dt represents time differentiation in axes that translate
and rotate with the mass element, and the r^’s are elements of a
nonsymmetric stress tensor that satisfies the relations
T32 “ r23 = (M X H)i «= M2H2 — • • •.
(8-47)
The equilibrium stress conditions are: in r,
d rt ^T21 дтз!
— (ru + 2тМ!2) + — + —- + M-VH! + Fy = 0, • • •;
dzi дхъ dx^
(8-48)
on S,
[th + 2ir(Mi2 - Mn2)]h + 721Z2 + T31Z3 = Tlt •... (8-49)
The formula for the work differs in several respects from the Voigt
formula, though it reduces to it in the two cases M = 0 and ец — 0.
Application of these formulas to specific problems leads to mathe-
matical complexities (Vlasov, 1960); consequently, not many such
problems have been solved, and none that are relevant to our
present topic of micromagnetics. Whether, with use of these rig-
orous formulas, magnetostriction could resolve the coercivity
paradox, is a question that cannot be answered at present.
Toupin (1956) has derived equations equivalent to equations
(8-47)-(8-49) in general systems of coordinates. He finds (in the
electric case) that if the stress and field components are assumed to
be linear functions of the displacement gradients and the polariza-
tion components, the resulting thepry is identical with Voigt’s
linear theory of piezoelectricity; but that when the relations are
assumed to be linear only in the displacement gradients and not in
the polarization components, the correct relations contain the rota-
tions as well as the strains. We must conclude that Voigt’s approxi-
mation was adequate for Voigt’s own purposes but is not adequate
for the uses since made of it in magnetostriction theory.
8.6 The Form Effect
An obvious error of the Voigt approximation was pointed out in
Section 8.1: by assuming an interaction energy in the form of the
volume integral of an energy density and neglecting the effect of
strains on the magnetostatic energy C7m, it fails to predict a shape
dependence of magnetostriction. The physical origin of this de-
pendence can be discussed more accurately at the microscopic
level. In fact, it was first noticed in attempts to calculate the mag-
netostrictive energy by evaluation of lattice sums. Since the mutual
energy of two dipoles depends on their mutual distance and on the
orientation of their moments (here assumed parallel), the total
dipole-dipole energy of a crystal is a function of the strains (as-
sumed uniform) and of the magnetization direction. For the un-
strained crystal this energy depends also on the shape of the
bounding surface, and this turns ^ut to be true likewise of the term
linear in the strains. Failure to realize this fact led to an apparent
discrepancy between results of Akulov (1928) and Becker (1930),
but this was quickly clarified by Powell (1931) and Hayasi (1931,
1934).
The dipole energy alone is too small to account for the main
part of the magnetostriction, but the additional forces that must
be invoked (spin-orbit, for example) are short-range and therefore
have no shape dependence. The form effect is thus entirely magne-
tostatic and long-range, and it can be handled by megascopic
theory, without use of lattice sums. This was done by Becker
(1934), and his predictions were verified by Kornetzki (1934).
Becker assumed a uniformly magnetized and strained ellipsoid;
the magnetostriction form effect then is due to the fact that the
magnetostatic energy Am, equation (3-28), depends on the strains.
The formulas for it involve the derivatives of the demagnetizing
factors with respect to the strains.
The form effect in a uniformly magnetized ellipsoid has been in-
vestigated in more detail by use of the formulas quoted in Section
8.5 (Brown, 1953). When this method is used, it turns out that the
strains are not uniform. The volume-average strains, however, are
those given by Becker’s formulas. Calculation of the displacements
as functions of position is difficult except for the sphere (Gersdorf,
1959).
Unless the applied field is large, it is not to be expected that the
magnetization will actually be uniform. One is then faced with the
necessity for calculating simultaneously the magnetization and the
strains. This might be a tractable problem at fields high enough to
justify linearization of the equations with respect to a and /3, as in
Section 4.6 and Chapter 5; but it has not been examined. Neither
has the problem of calculating nucleation fields with magnetostric-
tion taken into account and without use of the Voigt approxima-
tion.
Altogether, the development of a self-consistent theory that
treats magnetostriction without use of the Voigt approximation,
and magnetic structure without prior injection of the domain con-
cept, is very largely a problem for the future.
Symbols
a a, 6, • • • A Lattice constant. Constants. Helmholtz function: A = U — TS'. Vector potential.
b В c cij c ds dS dr 8 F Radius of cylinder or sphere. Magnetic flux density: В = H + 4irM. Speed of light. Elastic constants. Coi^tant in the exchange-energy integral. Line element. Surface element. Volume element. Strain component. Induced electromotive force. Dissipation function; occasionally, anisotropy-energy density.
Fi Sir Qij Component of body force per unit volume. Dissipation function per unit volume. Coefficients in the expansion of the anisotropy-energy density in powers of a and /3.
G G h II H' Gibbs function: G = A — jM-IIodr. Angular momentum per unit volume. Microscopic magnetic field intensity. Magnetizing force. The part of H due to the magnetization of the body under investigation.
Ho Applied field intensity, i.e., the part of H due to sources other than the magnetized body.
He H'® 3C », j> к Critical magnetizing force for instability. The parts of H' of orders 0, 1, 2 respectively in a and fl. “Effective” field intensity. Unit vectors along Cartesian coordinate axes.
J Constant in the exchange energy of two spins.
к ki, k2 KlfK2 lf m, n £ m M M, n A P,Q Boltzmann’s constant. Magnetostriction-energy constants. Anisotropy constants. Direction cosines of n. Lagrangian function. Torque (usually) per unit volume. Magnetic moment. Magnetization, i.e., magnetic moment per unit volume. Spontaneous magnetization. Unit outward normal. Demagnetizing factor; (occasionally) integer. Integrals such that P + MeHQ is the change in Gt to the second order in a and в, from a reference state a = /3 = 0.
r r r° s Magnitude of r. Position vector. Unit vector along r. Position vector of a nearest-neighbor spin site with respect to the site being considered.
s s Elastic moduli: (e#) = (c*/)”1. Spin angular momentum. Surface; magnitude of spin angular momentum.
S' t T Ti 5 11 U V y, z «1, «2, «3 «, У 70 7u’ <5 50 Entropy (usually) per unit volume. Time. Kelvin temperature; kinetic energy. Component of surface force per unit area. Kinetic energy per unit volume. Component of elastic displacement. ai + ftj. Internal energy; scalar potential. Unit vector along M: v = ai + 0j + 7k» Cartesian components of a point. = ft 7. Direction cosines of M. Ratio of magnetic moment to angular momentum. Domain wall energy per unit area. The tensor (дц) (i — 1, 2). Small change; domain wall thickness. 1 if j = г; 0 if j 5* t. Small vector rotation (= <06/ for an actual rotation in time interval 60.
d/dv d/dv id/da + jd/dfi + kd/d7. id/da + jd/dfi + kd/d-y.
Dissipation constant. the z component of magnetizing force, in energy- density units, in the reference state M = Л/Зк.
X D, ^111 Lagrangian multipliers; in nucleation theory, X = — к. Magnetostriction constants. Volume.
। т; Stress component. Magnetic flux.
О Angles describing the orientation of M: a = sin 0 cos Ф, 3 — sin О sin Ф, у — cos O. Angular frequency, i.e., frequency times 2r. Angular velocity.
Д2
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Index
Abraham, C., 71, 94
Aharoni, A., 71, 72, 78, 79, 80, 94, 97,
112
Akulov, N. S., 11, 14, 15, 130
Angular momentum, 24
Anisotropy, crystalline (magnetocrys-
talline), 8, 42, 69; see also
Energy, anisotropy
from lattice sums, 11
phenomenological theory, 34
surface, 37, 42, 69
Approximate calculations (methods),
82, 98
Bean, С. P., 71
Becker, R., 2,11,12,14,18,19,64,130
Bitter, F., 16, 85, 91
Bloch, F., 15, 68
Boundary-value problem, 48, 74
Bozorth, R. M., 68
Brown, W. F., Jr., 15, 16, 18, 20, 22,
38, 41, 44, 63, 64, 66, 69, 70, 71,
77, 80, 82, 83, 84, 90, 97, 99,
105, 114, 128, 130
Cavities, 66
Coercivity, 65
Coercivity paradox, 66, 70, 94
effect of magnetostriction on, 126
Courant, R., 52, 73
Critical diameter (radius, size), 22, 77,
80
Critical diameter (radius, size), effect
of magnetostriction on, 127
Crystals, cubic, critical magnetizing
force for instability, 67, 68
forms of energy terms, 8, 34, 35
lattice sums, 11
hexagonal, 8, 34, 36
Cylinder, finite, 100
infinite, magnetization curve, 96
nucleation fields and modes, 76,
80
resonance fields and modes, 112
Damping; see Dissipation
De Blois, R. W., 71
Demagnetizing factor, of ellipsoid, 13,
67, 75 [41
of any uniformly magnetized body,
Direction angles of magnetization, 2, 4
Direction cosines of magnetization, 2,
4
Direction of easy magnetization,
66-67, 80
Discontinuous solutions, 92
Dislocations, 18, 64, 66
Dissipation (damping), in dynamic
test of stability, 54
effect on precession, 32, 103, 105
in equation of motion, 6, 51, 115
phenomenological theory, 43
Distribution, of orientations, 65, 82,
108
Distribution, of shapes, 82, 105, 108
of sizes, 82
Domain structure, 92, 101
Domain theory, 10, 17, 99
primitive, 12
Domains, 2, 4, 12
Dorfman, J., 14, 19
D6ring, W., 2, 14, 18, 19
Dynamic, criterion for static stability,
54, 105
effects, 4,*JF
equation, 51
problems, 101
Eddy currents, 52
Elastic theory, 117, 121
Ellipsoid, crystal specimen, 67, 68
magnetostriction form effect, 130
nucleation field, 75, 80
replacement of parallelepiped by, 99
as size becomes infinite, 86
uniform mode of oscillation, 102
uniformly magnetized, 41, 55, 62
Elmore, W. C., 16, 18
Energy, anisotropy, definition, 8
in domain structure, 92
field to overcome, 76
in one-dimensional problems, 85,
88
methods of calculating, 9, 11, 34
minimum of, 67
distinguished from free energy, 6
elastic, 8, 120
exchange, 8
in domain structure, 92
field to overcome, 76
finiteness in magnetization curl-
ing, 97
methods of calculating, 35, 37
in one-dimensional calculations,
85
variation of, 42
free, 6, 28
Energy, free, formulas for, 32,120; see
also Helmholtz function, Gibbs
function
functions, methods of evaluating, 9
internal, 5, 116
magnetic (magnetostatic), as com-
plicating factor, 17, 65
definition, 8
in domain-wall theory, 66, 91
field to overcome, 77
method of calculating, 32
neglect of, 12
in one-dimensional calculations,
86
transformations of, 38, 42
upper and lower bounds, 83
magnetostrictive, 8, 120 ff.
potential, 5-0, 29, 32
surface, 36
Entropy, 6, 28
Equation of motion (dynamic torque
equation), 25, 51
Gilbert and Landau-Lifshitz forms,
44
Lagrangian form, 45
linearized, 54, 60
Equilibrium equations, elastic, 121,
129
magnetic, of cylinder, 97
in one-dimensional problems, 85
in reference state, 58, 69
torque condition, 4, 24, 41
of uniformly magnetized particle,
22, 54
from variational principle, 26, 29,
46
Field, alternating, 53, 58, 101
applied, 27, 57, 58
effective, 6, 25, 48
Lorentz, 33, 36
molecular, 1
nucleation; see Nucleation field
Films, 70, 80, 84
Fine, P. C., 28
Forces, anisotropy, 12, 101; see also
Energy, anisotropy
dipole-dipole, 3, 7, 12; see also
Energy, magnetic
in one-dimensional problems,
86
exchange, 1, 12, 101; see also
Energy, exchange
magnetic; see Energy, magnetic
magnetostrictive, 3; see also Energy,
magnetostrictive
Form effect, magnetostriction, 128,
130
Fourier methods, 64, 66, 99
Free energy; see Energy, free
Frei, E. H., 22
Frenkel, J., 14, 19
Gans, R., 12
Gersdorf, R., 131
Gibbs function G, 28, 121
Gilbert, T. L., 44, 45
Goldstein, H., 31
Goodenough, J. B., 71
Grimes, D. M., 15
Guillaud, C., 20
Gyorgy, E. M., 115
Halverson, R. P., 100
Hamilton’s integral (principle), 31,
42, 122
in dynamic analog to one-dimen-
sional equilibrium problem, 89
Hanton, J. P., 105
Hayasi, T., 130
Heisenberg, W., 14
Helmholtz function A, 5, 8, 28
calculation of, 32, 120, 123
Hilbert, D., 73
Historical background, 11
Holstein, T., 16, 66
Huber, E. E., Jr., 80
Hysteresis, 21, 29
Imperfections, 71, 72, 115
specific types of, 95
Interaction (magnetic) of particles, 82
magnetomechanical (magnetoelas-
tic), 118, 122; see also Energy,
magnetostrictive
Inversion, 11, 18, 68
Jeffreys, B., 64
Jeffreys, H., 64
Kersten, M., 14
Kittel, C., 19, 20, 21, 102, 103
Kondorskii, E., 22, 98
Kometzki, M., 130
Kronmiiller, H., 65
Lagrangian, function, 31
multipliers, 26, 41, 49
Landau, L., 15, 18, 19, 44, 68
Landau-Lifshitz theory, of domain
structure, 18, 65
of wall, 15, 16, 68
Lifshitz, E., 15, 18, 19, 44, 68
Linearized equations, 54, 55, 61
in dynamic problems, 101
in magnetostriction, 131
Lorentz, H. A., 33
Love, A. E. H., 38, 117, 121
Magnetization buckling, 78, 80
Magnetization curling, in cylinder,
76, 97, 115
dynamic generalization, 112, 114
effect of magnetostriction on, 127
in prolate spheroid, 80
in sphere, 78
Magnetization (M vs H, M vs Ho)
curve, 50, 70, 100
slope of, 50, 70
Magnetization, spontaneous, 1
Magnetostatic modes, 106
Magnetostatic theorems, 38
Magnetostriction, 14, 116; see also
Energy, magnetostrictive
Megascopic, 13
Micromagnetics, early calculations in,
16
recent calculations in, 21
Minimization principles, thermostatic,
26
Modes of deviation, 75
Modes of motion (oscillation, pre-
cession), 54, 101
Morrish, A. H., 41, 105
Muller, M. W., 82, 98
Ndel, L., 17, 21, 22, 37, 66
Nonlinear calculations, 85, 115
Nucleation of domains, 71, 72, 126
Nucleation field, 72, 75, 82
effect of magnetostriction on, 127
and resonance field, 107
One-dimensional cases, 85
Orthogonality, 109, 111
Pole-avoidance principle, 17, 39
in domain theory, 65, 92, 101
Poles, 83, 87; see also Pole-avoidance
principle
Polycrystalline specimen, 65
Powder, 82, 104
Powell, F. C., 130
Precession, 25, 32, 54
Primakoff, H., 17, 66
Propagation speed, 52
Random deviating forces, 63, 64
Rayleigh-Ritz (Ritz) method, 98, 99,
100
Resonance, 4, 10, 101
Resonance fields and nucleation fields,
107
Ritz (Rayleigh-Ritz) method, 98, 99,
100
Rotation in unison, 70, 75
in cylinder, 76, 97, 100
dynamic generalization, 102, 112,
114
effect of magnetostriction on, 127
Rotation theory, of Akulov, 11, 66, 75
of Becker, 14, 18
Panofsky, W. К. H., 27
Paradox, coercivity, 66, 70, 94
effect of magnetostriction on,
126
Particle, fine, 2, 70, 72; see also Par-
ticle, single-domain
in alternating field, 101
behavior throughout field cycle,
80
single-domain, 19, 29, 75
in alternating field, 101, 104
throughout field cycle, 80
Periodic boundary conditions, 106
Phillips, M., 27
Physically small, 33
Plate, 80, 98
Saturation, approach to, 61
Seavey, M. H., Jr., 102
Seeger, A., 65
Shape dependence of magnetic energy,
86, 130
Shape distribution; see Distribution
Shirobokov, M., 16, 85
Shtrikman, S., 22, 71, 78, 79, 80, 90,
97, 112
Sixtus, K. J., 14, 15
Smith, D. O., 80
Snoek, J. L., 103
Sokolnikoff, I. S., 117
Soohoo, R. F., 104, 106
Sphere, 78, 97
magnetostriction form effect, 131
Spheroid (ellipsoid of revolution), 67,
75, 80
Spin wave, 10, 106
Stability, condition for, 24, 26, 29
dynamic criterion, 54, 105, 108
in infinite cylinder, 96
in one-dimensional problems, 85, 90
in rotation theory, 67
by second variation, 49
of uniform magnetization, 69, 71, 82
Statistical theory, 15, 44, 72
Steady-state problems (alternating
field), 53
Stoner, E. C., 21, 71, 97
Strains, 8, 117 ff.
Stratton, J. A., 78
Stress, internal, 12, 18, 65
Stresses, 8, 81, 118 ff.
Susceptibility, apparent, 104
Tannenwald, P. E., 102
Temperature, 4-6, 28
Thermal agitation (fluctuations), 44,
71
Thermodynamic (thermostatic) the-
ory, 26, 116
Tolman, R. C., 28
Tonks, L., 14, 15
Torque, 4
Torque, equation, equilibrium, 46
dynamic, 51
equations, 24, 41
Toupin, R. A., 129
Transient phenomena (problems), 5,
53, 114
Treves, D., 22, 71, 80
Uniform mode of oscillation (pre-
cession), 102
Variational principles, dynamic, 30,
44, 51
procedure, 5, 46, 59
techniques, 41
Vicena, F., 66
Vlasov, К. B., 129
Voigt approximation, 116, 128
Voigt, W., 116
Walker, L. R., 106
Wall Displacement, 18, 68
Wall, Interdomain (Bloch), 15, 68, 91
Wehlau, A., 98
Weiss-Heisenberg theory, 1, 3
Whisker, 71, 72, 84
White, R. L., 106
Whittaker, E. T., 31
Wohlfarth, E. P., 21, 97