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`coherence phenomena in electron-hole and coupled matter-light systems
`I. M. Vardavas, F. W. Taylor: Radiation and climate
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`G. C. Branco, L. Lavoura, J. P. Silva: CP violation
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`L. M. Pismen: Vortices in nonlinear fields
`L. Mestel: Stellar magnetism
`K. H. Bennemann: Nonlinear optics in metals
`M. Brambilla: Kinetic theory of plasma waves
`S. Chikazumi: Physics of ferromagnetism
`R. A. Bertlmann: Anomalies in quantum field theory
`P. K. Gosh: Ion traps
`S. L. Adler: Quaternionic quantum mechanics and quantum fields
`P. S. Joshi: Global aspects in gravitation and cosmology
`E. R. Pike, S. Sarkar: The quantum theory of radiation
`P. G. de Gennes, J. Prost: The physics of liquid crystals
`M. Doi, S. F. Edwards: The theory of polymer dynamics
`S. Chandrasekhar: The mathematical theory of black holes
`C. M(Zil!er: The theory of relativity
`H. E. Stanley: Introduction to phase transitions and critical phenomena
`A. Abragam: Principles of nuclear magnetism
`P. A. M. Dirac: Principles of quantum mechanics
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`- - - - - - - - - - - - - - - - - - - - - - - - - -
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`- - - - - - - - - -~------- - -
`
`Physics of
`Ferromagnetism
`
`SECOND EDITION
`
`~ SOSHIN CHIKAZUMI
`Professor Emeritus,
`University of Tokyo
`
`English edition prepared with the assistance of
`C. D. GRAHAM, JR
`Professor Emeritus,
`University of Pennsylvania
`
`OXFORD
`
`UNIVERSITY PRESS
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`OXFORD
`UNIVERSITY PlUlSS
`Great Clarendon Street, Oxford oxz 6DP
`Oxford University Press is a department of the University of Oxford.
`It furthers the University's objective of excellence in research, scholarship,
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`Oxford is a registered trade mark of Oxford University Press
`in the UK and in certain other countries
`Published in the United States
`by Oxford University Press Inc., New York
`© Siishin Chikazmpi, 1997
`The moral rights of the author have been asserted
`Database right Oxford University Press (maker)
`In 1978 and 1984 substantial portions of this book were published
`in Japanese in two volumes by Syokabo Publishing Company, Tokyo,
`as a revised edition of the first edition published in 1959 by the same
`company. The English version of the first edition was published by
`John Wiley & Sons in 1964 under the title Physics of Magnetism.
`First published by Oxford University Press, 1997
`First published in paperback 2009
`All rights reserved. No part of this publication may be reproduced,
`stored in a retrieval system, or transmitted, in any form or by any means,
`without the prior permission in writing of Oxford University Press,
`or as expressly permitted by law, or under terms agreed with the appropriate
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`outside the scope of the above should be sent to the Rights Department,
`Oxford University Press, at the address above
`You must not circulate this book in any other binding or cover
`and you must impose this same condition on any acquirer
`A catalogue record for this book is available from the British Library
`Library of Congress Cataloging in Publication Data
`Chikazumi, Siishin, 1922-
`Physics of ferromagnetism f Siishin Chikazumi; English ed.
`prepared with the assistance of C. D. Graham, Jr. - 2nd ed.
`English version of first ed. published under title: Physics of
`magnetism. New York : Wiley, 1964.
`Includes bibliographical references and indexes.
`1. Ferromagnetism. I. Graham, C. D. (Chad D.) II. Chil,azumi,
`Siishin, 1922- Physical of magnetism. III. Title.
`538'.44-dc20
`96-27148 CIP
`QC761.4.C47 1997
`ISBN 978-0-19-851776-4 (Hbk)
`ISBN 978-0-19-956481-1 (Pbk)
`
`Printed in Great Britain
`by
`CPI Antony Rowe, Chippemnham, Wiltshire
`
`LMBTH-000009
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`

`
`PREFACE
`
`This book is intended as a textbook for students and investigators interested in the
`physical aspect of ferromagnetism. The level of presentation assumes only a basic
`knowledge of electromagnetic theory and atomic physics and a general familiarity
`with rather elementary mathematics. Throughout the book the emphasis is primarily
`on explanation of physical concepts rather than on rigorous theoretical treatments
`which require a background in quantum mechanics and high-level mathematics.
`Ferromagnetism signifies in its wide sense the strong magnetism of attracting pieces
`of iron and has long been used for motors, generators, transformers, permanent
`magnets, magnetic tapes and disks. On the other hand, the physics of ferromagnetism
`is deeply concerned with quantum-mechanical aspects of materials, such as the
`exchange interaction and band structure of metals. Between these extreme limits,
`there is an intermediate field treating magnetic anisotropy, magnetostriction, domain
`structures and technical magnetization. In addition, in order to understand the
`magnetic behavior of magnetic materials, we need some knowledge of chemistry and
`crystallography.
`The purpose of this book is to give a general view of these magnetic phenomena,
`focusing its main interest at the center of this broad field. The book is divided into
`eight parts. After an introductory description of magnetic phenomena and magnetic
`measurements in Part I, the magnetism of atoms including nuclear magnetism and
`microscopic experiments on magnetism, such as neutron diffraction and nuclear
`magnetic resonance (NMR), is treated in Part II. The origin and mechanism of para-,
`ferro- and ferrimagnetism are treated in Part III. Part IV is devoted to more
`material-oriented aspects of magnetism, such as magnetism of metals, oxides, com-
`pounds and amorphous materials. In Part V, we discuss magnetic anisotropy and
`magnetostriction, to which I have devoted most of my research life. Part VI describes
`domain structures, their observation technique and domain theory. Part VII is on
`magnetization processes, analyzed on the basis of domain theory. Part VIII is devoted
`to phenomena associated with magnetization such as magnetothermal, magnetoelec-
`trical and magneto-optical effects, and to engineering applications of magnetism.
`Throughout the book, the SI or MKSA system of units using the E-H analogy is
`used. As is well known, this system is very convenient for describing all electromag-
`netic phenomena without introducing troublesome coefficients. This system also uses
`practical units of electricity such as amperes, volts, ohms, coulombs and farads. This
`system is particularly convenient when we treat phenomena such as eddy currents and
`electromagnetic induction which relate magnetism to electricity. However, old-timers
`who are familiar with the old CGS magnetic units such as gauss, oersted, etc., must
`change their thinking from these old units to new units such as tesla, ampere per
`meter, etc. Once they become familiar with the new magnetic unit system, however,
`they may come to appreciate its convenience. To aid in the transition, a conversion
`table between MKSA and CGS units is given in Appendix 5.
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`Vl
`
`PREFACE
`In the previous edition I tried to refer to as many papers as possible. By the time of
`the revised edition, so many papers had become available that I was obliged to select
`only a small number of them to keep the text clear and simple. I have no doubt
`omitted many important papers for this reason, for which I apologize and beg their
`authors for tolerance and forgiveness. Many authors have kindly permitted me to use
`their beautiful photographs and unpublished data, for which I want to express my
`sincere thanks.
`This book was originally published in Japanese by Shyokabo Publishing Company in
`Tokyo in 1959. The English version of that edition was published by John Wiley &
`Sons, Inc. in New York in 1964. The content of the English version was increased by
`about 55% from the Japanese version. At that time my English was polished by Dr
`Stanley H. Charap. A revised Japanese edition was published in two volumes in 1978
`and 1984, respectively. The content was about 30% larger than the previous English
`edition. The preparation of the present English version of the revised edition was
`started in 1985 and took about ten years. This time my English was polished by
`Professor C. D. Graham, Jr using e-mail communication. The content has not been
`greatly increased, but has been renewed by introducing recent developments and
`omitting some old and less useful material.
`Thanks are due to the staff of Oxford University Press who have helped and
`encouraged me throughout the period of translation.
`
`Tokyo
`March 1996
`
`S.C.
`
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`

`
`CONTENTS
`
`Part I Classical Magnetism
`
`1 MAGNETOSTATIC PHENOMENA
`1.1 Magnetic moment
`1.2 Magnetic materials and magnetization
`1.3 Magnetization of ferromagnetic materials and demagnetizing fields
`1.4 Magnetic circuit
`1.5 Magnetostatic energy
`1.6 Magnetic hysteresis
`Problems
`References
`
`2 MAGNETIC MEASUREMENTS
`2.1 Production of magnetic fields
`2.2 Measurement of magnetic fields
`2.3 Measurement of magnetization
`Problems
`References
`
`Part II Magnetism of Atoms
`
`3 ATOMIC MAGNETIC MOMENTS
`3.1 Structure of atoms
`3.2 Vector model
`3.3 Gyromagnetic effect and ferromagnetic resonance
`3.4 Crystalline field and quenching of orbital angular momentum
`Problems
`References
`
`4 MICROSCOPIC EXPERIMENTAL TECHNIQUES
`4.1 Nuclear magnetic moments and related experimental techniques
`4.2 Neutron diffraction
`4.3 Muon spin rotation ( p.SR)
`Problems
`References
`
`3
`3
`7
`11
`17
`22
`27
`31
`32
`
`33
`33
`39
`42
`49
`49
`
`53
`53
`59
`68
`74
`81
`83
`
`84
`84
`93
`100
`104
`104
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`viii
`
`CONTENTS
`Part III Magnetic Ordering
`5 MAGNETIC DISORDER
`5.1 Diamagnetism
`5.2 Paramagnetism
`Problems
`References
`
`6 FERROMAGNETISM
`6.1 Weiss theory of ferromagnetism
`6.2 Various statistical theories
`6.3 Exchange interaction
`Problems
`References
`
`7 ANTIFERROMAGNETISM AND FERRIMAGNETISM
`7.1 Antiferromagnetism
`7.2 Ferrimagnetism
`7.3 Helimagnetism
`7.4 Parasitic ferromagnetism
`7.5 Mictomagnetism and spin glasses
`Problems
`References
`
`Part IV Magnetic Behavior and Structure of Materials
`8 MAGNETISM OF METALS AND ALLOYS
`8.1 Band structure of metals and their magnetic behavior
`8.2 Magnetism of 3d transition metals and alloys
`8.3 Magnetism of rare earth metals
`8.4 Magnetism of intermetallic compounds
`Problems
`References
`
`9 MAGNETISM OF FERRIMAGNETIC OXIDES
`9.1 · Crystal and magnetic structure of oxides
`9.2 Magnetism of spinel-type oxides
`9.3 Magnetism of rare earth iron garnets
`9.4 Magnetism of hexagonal magnetoplumbite-type oxides
`9.5 Magnetism of other magnetic oxides
`Problems
`References
`
`107
`107
`110
`116
`117
`
`118
`118
`124
`129
`133
`133
`
`134
`134
`142
`148
`151
`153
`158
`159
`
`163
`163
`173
`181
`188
`193
`193
`
`197
`197
`199
`207
`210
`215
`220
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`CONTENTS
`10 MAGNETISM OF COMPOUNDS
`10.1 3d Transition versus Illb Group magnetic compounds
`10.2 3d-1Vb Group magnetic compounds
`10.3 3d-Vb Group magnetic compounds
`10.4 3d-Vlb Group magnetic compounds
`10.5 3d-Vllb (halogen) Group magnetic compounds
`10.6 Rare earth compounds
`Problems
`References
`
`11 MAGNETISM OF AMORPHOUS MATERIALS
`11.1 Magnetism of 3d transition metal-base amorphous materials
`11.2 Magnetism of 3d transition plus rare earth amorphous alloys
`Problem
`References
`
`Part V Magnetic Anisotropy and Magnetostriction
`12 MAGNETOCRYSTALLINE ANISOTROPY
`12.1 Phenemenology of magnetocrystalline anisotropy
`12.2 Methods for measuring magnetic anisotropy
`12.3 Mechanism of magnetic anisotropy
`12.4 Experimental data
`Problems
`References
`
`13 INDUCED MAGNETIC ANISOTROPY
`13.1 Magnetic annealing effect
`13.2 Roll magnetic anisotropy
`13.3 Induced magnetic anisotropy associated with crystallographic
`transformations
`13.4 Other induced magnetic anisotropies
`Problems
`References
`
`14 MAGNETOSTRICTION
`14.1 Phenomenology of magnetostriction
`14.2 Mechanism of magnetostriction
`14.3 Measuring technique
`14.4 Experimental data
`14.5 Volume magnetostriction and anomalous thermal expansion
`14.6 Magnetic anisotropy caused by magnetostriction
`
`ix
`222
`223
`226
`228
`232
`234
`235
`236
`236
`
`239
`240
`243
`244
`245
`
`249
`249
`256
`266
`274
`296
`296
`
`299
`299
`309
`
`318
`329
`339
`339
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`343
`343
`349
`357
`359
`363
`376
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`
`CONTENTS
`14.7 Elastic anomaly and magnetostriction
`Problems
`References
`
`Part VI Domain Structures
`15 OBSERVATION OF DOMAIN STRUCTURES
`15.1 History of domain observations and powder-pattern method
`15.2 Magneto-optical method
`15.3 Lorentz electron microscopy
`15.4 Scanning electron microscopy
`15.5 X-ray topography
`15.6 Electron holography
`References
`
`16 SPIN DISTRIBUTION AND DOMAIN WALLS
`16.1 Micromagnetics
`16.2 Domain walls
`16.3 180° walls
`16.4 90° walls
`16.5 Special-type domain walls
`Problems
`References
`
`17 MAGNETIC DOMAIN STRUCTURES
`17.1 Magnetostatic energy of domain structures
`17.2 Size of magnetic domains
`17.3 Bubble domains
`17.4 Stripe domains
`17.5 Domain structure of fine particles
`17.6 Domain structures in non-ideal ferromagnets
`Problems
`References
`
`Part VII Magnetization Processes
`18 TECHNICAL MAGNETIZATION
`18.1 Magnetization curve and domain distribution
`18.2 Domain wall displacement
`18.3 Magnetization rotation
`18.4 Rayleigh loop
`18.5 Law of approach to saturation
`
`379
`381
`381
`
`387
`387
`393
`394
`396
`401
`402
`405
`
`407
`407
`411
`417
`422
`428
`432
`432
`
`433
`433
`439
`445
`450
`453
`457
`463
`464
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`467
`467
`480
`491
`498
`503
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`--------------------------------
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`CONTENTS
`
`18.6 Shape of hysteresis loop
`Problems
`References
`
`19 SPIN PHASE TRANSITION
`19.1 Metamagnetic magnetization processes
`19.2 Spin flop in ferrimagnetism
`19.3 High-field magnetization process
`19.4 Spin reorientation
`Problems
`References
`
`20 DYNAMIC MAGNETIZATION PROCESSES
`20.1 Magnetic after-effect
`20.2 Eddy current loss
`20.3 High-frequency characteristics of magnetization
`20.4 Spin dynamics
`20.5 Ferro-, ferri-, and antiferro-magnetic resonance
`20.6 Equation of motion for domain walls
`Problems
`References
`
`Part VIII Associated Phenomena
`and Engineering Applications
`21 VARIOUS PHENOMENA ASSOCIATED WITH
`MAGNETIZATION
`21.1 Magnetothermal effects
`21.2 Magnetoelectric effects
`21.3 Magneto-optical phenomena
`References
`
`22 ENGINEERING APPLICATIONS OF MAGNETIC
`MATERIALS
`22.1 Soft magnetic materials
`22.2 Hard magnetic materials
`22.3 Magnetic memory and memory materials
`References
`
`Solutions to problems
`
`Appendix 1. Symbols used in the text
`
`xi
`509
`516
`516
`
`518
`518
`521
`529
`533
`535
`536
`
`537
`537
`551
`556
`562
`567
`574
`580
`581
`
`585
`585
`590
`596
`598
`
`600
`600
`605
`608
`613
`
`615
`
`628
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`
`xii
`CONTENTS
`Appendix 2. Conversion of various units of energy
`
`Appendix 3.
`
`Important physical constants
`
`Appendix 4. Periodic table of elements and magnetic elements
`
`Appendix 5. Conversion of magnetic quantities- MKSA and CGS systems
`
`Appendix 6. Conversion of various units for magnetic field
`
`Material index
`
`Subject index
`
`631
`
`632
`
`633
`
`638
`
`639
`
`641
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`12
`MAGNETOCRYSTALLINE ANISOTROPY
`
`12.1 PHENOMENOLOGY OF MAGNETOCRYSTALLINE
`ANISOTROPY
`
`The term magnetic anisotropy is used to describe the dependence of the internal
`energy on the direction of spontaneous magnetization. We call an energy term of this
`kind a magnetic anisotropy energy. Generally the magnetic anisotropy energy term has
`the same symmetry as the crystal structure of the material, and we call it a
`magnetocrystalline anisotropy.
`The simplest case is uniaxial magnetic anisotropy. For example, hexagonal cobalt
`exhibits uniaxial anisotropy with the stable direction of spontaneous magnetization, or
`easy axis, parallel to the c-axis of the crystal at room temperature. As the magnetiza-
`tion rotates away from the c-axis, the anisotropy energy initially increases with 0, the
`angle between the c-axis and the magnetization vector, then reaches a maximum
`value at e = 90°, and decreases to its original value at (} = 180°. In other words, the
`anisotropy energy is minimum when the magnetization points in either the + or -
`direction along the c-axis. We can express this energy by expanding it in a series of
`powers of sin 2 8:
`
`(12.1)
`where cp is the azimuthal angle of the magnetization in the plane perpendicular to the
`c-axis. Using the relationships
`cos 2 (} = }(1 + cos28),
`sin 2 (} = } (1 - cos 2 (}),
`(12.1) is converted to a series in cos nO (n = 2, 4, 6, ... ) as
`Ea = }Ku1(1- cos28) + iKu2(3- 4cos2(} + cos48)
`+ tzKu 3(10- 15 cos 28 + 6 cos4(}- cos 60)
`+ tzKu4(10- 15 COS 2(} + 6 COS 4(}- COS 6(}) COS 6cp + • · • .
`(12.2)
`The coefficients Kun (n = 1, 2, ... ) in these equations are called anisotropy constants.
`The values of uniaxial anisotropy constants of cobalt at 15°C1 are
`Ku 1 = 4.53 X 10 5 J m- 3 ( = 4.53 X 106 ergcm- 3
`)).
`Kuz = 1.44 X 105 Jm- 3 ( = 1.44 X 106 ergcm- 3 )
`
`(12.3)
`
`The higher-order terms are small and their values are not reliably known. If these
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`250
`
`MAGNETOCRYSTALLINE ANISOTROPY
`z
`
`X
`Fig. 12.1. Equivalent directions in a cubic crystal.
`
`constants are positive as in the case of cobalt, the anisotropy energy, E., increases
`with increasing angle 8, so that E is minimum at () = 0. In other words, the
`spontaneous magnetization is stable when it is parallel to the c-axis. Such an axis is
`called an axis of easy magnetization or simply an easy axis. If these constants are
`negative, the anisotropy energy is maximum at the c-axis, so that it becomes unstable.
`Such an axis is called an axis of hard magnetization or simply a hard axis.
`In the latter case, the magnetization is stable when it lies in any direction in the
`c-plane (8 = 90°). Such a plane is called a plane of easy magnetization or an easy plane.
`If Ku1 > 0 and Ku2 < 0, the stable direction of magnetization forms a cone, which is
`called a cone of easy magnetization or an easy cone.
`For cubic crystals such as iron and nickel, the anisotropy energy can be expressed in
`terms ofthe direction cosines (a1, a 2 , a 3) of the magnetization vector with respect to
`the three cube edges. There are many equivalent directions in which the anisotropy
`energy has the same value, as shown by the points A 1, A 2 , B1, B2 , Cl> and C 2 on an
`octant of the unit sphere in Fig. 12.1. Because of the high symmetry of the cubic
`crystal, the anisotropy energy can be expressed in a fairly simple way: We expand the
`anisotropy energy in a polynomial series in a1, a2 , and a3 • Those terms which
`include the odd powers of a; must vanish, because a change in sign of any of the a;
`should bring the magnetization vector to a direction which is equivalent to the
`original direction. The expression must also be invariant to the interchange of any two
`a;s so that the terms of the form a?hfmazn must have, for any combination of
`1, m, n, the same coefficient for any interchange of i, j, k. The first term, therefore,
`should have the form af + ai + aff, which is always equal to 1. Next is the fourth-order
`term which can be reduced to the form I:;> 1 a}af by the relationship
`a{+ ai + aj = 1- 2(afai + aiaff + affat}.
`Thus we have the expression
`E. =K1(afai + aiaff + a~af) +K2 afa}aff
`2
`+ K3(afai + aJa~ + a~af) + ···,
`
`(12.4)
`
`(12.5)
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`PHENOMENOLOGY OF MAGNETOCRYSTALLINE ANISOTROPY 251
`z
`z
`
`I
`
`y
`
`y
`
`./
`
`X
`
`the cubic
`Fig. 12.2. Polar diagram of
`anisotropy energy for K 1 > 0 and K 2 = 0.
`(Radial vector is equal to E.+ tK1.)
`
`the cubic
`Fig. 12.3. Polar diagram of
`anisotropy energy for K1 < 0 and K 2 = 0.
`(Radial vector is equal to E.+ 2IK11.)
`
`)
`
`•
`
`(12.6)
`
`where K1, K 2 , and K 3 are the cubic anisotropy constants. For iron at 20°C,2
`K1 = 4.72 X 104 J m- 3 ( = 4.72 X 105 ergcm- 3
`)
`K 2 = -0.075X10 4 Jm- 3 (= -0.075x105 ergcm- 3 )
`and, for nickel at 23oC,3
`K1 = -5.7 X 103 J m- 3 ( = -5.7 X 104 ergcm- 3))
`K 2 = -2.3X10 3 Jm- 3 (= -2.3X104 ergcm- 3 )
`•
`For [100], a 1 = 1, a 2 = a 3 = 0, so that the value of E. given by (12.5) is
`E0 =0,
`and, for [111], a 1 = a 2 = a 3 = 1/ f3, so that
`E. = tK1 + f.TK 2 + ~K3 + .. · .
`(12.9)
`If K1 > 0 as in the case of iron, and ignoring the K 2 and K3 terms, E. for [111] is
`higher than that for [100], so that [100] becomes the easy axis. Considering the cubic
`symmetry, [010] and [001] are also easy axes. Figure 12.2 shows a polar diagram of the
`anisotropy energy in this case. This diagram is a locus of the vector drawn from the
`origin in the direction of the spontaneous magnetization with length equal to the
`anisotropy energy given by (12.5) plus a constant term equal to ~K1 • It is seen that the
`surface is concave, or in other words the energy is minimum, in the x-, y-, and z- (or
`[100], [010], and [001]) directions. If K1 < 0 as in the case of nickel, E. < 0 for [111] as
`we see in (12.9) (ignoring K 2 and K 3 terms), so that E. for [111] is lower than that
`for [100] and [111] and its equivalent [111], [111], and [111] directions are easy axes.
`Figure 12.3 shows a polar diagram of the anisotropy energy, plus a constant term
`2IK11, for K1 < 0 and K 2 =K3 = 0. It is seen that the surface is convex for cube axes
`
`(12.7)
`
`(12.8)
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`MAGNETOCRYSTALLINE ANISOTROPY
`z
`z
`
`y
`
`y
`
`Fig. 12.4. Polar diagram of the cubic
`anisotropy energy for K1 = 0 and K2 > 0.
`(Radial vector is equal to Ea + tK2 .)
`
`the cubic
`Fig. 12.5. Polar diagram of
`anisotropy energy for K1 = 0 and K2 < 0.
`(Radial vector is equal to Ea + fr,K2 .)
`
`and concave for the (111) axes. Figures 12.4 and 12.5 show the anisotropy energy
`surfaces for K 1 = K 3 = 0 and K 2 > 0 and for K 1 = K 3 = 0, and K 2 < 0, respectively. In
`these cases, the energy is equal to zero for all directions in the x-y, y-z, and z-x
`planes, but Ea for (111) is highest in the case of positive K 2 and lowest in the case of
`negative K 2• However, in many ferro- or ferrimagnetic materials K 1 > K 2 , and even if
`K 1 = K 2 , the change in a formula for the K 2 term is only i of that from the K 1 term,
`so that the contribution of the K 2 term can be ignored.
`When, however, the magnetization rotates in some particular crystallographic
`plane, the K 2 term is not necessarily negligible. This is the case if the magnetization
`is confined to the {111} plane.
`Before examining the {111} plane, let us consider the case in which the magnetiza-
`tion rotates in the x-y or (001) plane. If (} is the angle between the magnetization
`and the x-axis (see Fig. 12.6), then we have
`
`a: 1 = c~s (J}
`a:2 = sm (}
`0:3 = 0
`
`.
`
`(12.10)
`
`Using this relationship, we have from (12.5)
`
`K3
`Kl
`= -(1- cos4fJ) + -(3- 4cos4(J + cos8e) + ....
`8
`128
`
`(12.11)
`
`Note that there is no contribution from the K 2 term.
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`PHENOMENOLOGY OF MAGNETOCRYSTALLINE ANISOTROPY 253
`z
`
`y
`
`X
`Fig. 12.6. Rotation of magnetization in the
`(001) plane out of the x-direction.
`Next we consider the (llO) plane. If e is the angle of rotation of the magnetization
`from the z-axis in the (110) plane (see Fig. 12.7), then we have the relationship
`
`X
`Fig. 12.7. Rotation of magnetization in the
`(110) plane out of the z-direction.
`
`a 1 = a 2 = ~ sin e} .
`a 3 =cos e
`
`(12.12)
`
`Using this relationship, we have from (12.5) the anisotropy energy
`
`E. = K1( t sin4 e + sin2 e cos2 e) + tK2 sin4 e cos2 e
`+K3(t sin4 e + sin2 e cos2 e)2 + ...
`= f-IK1(7- 4cos20- 3 cos400) + 1~8 K2 (2- cos2e- 2cos4e
`+cos60) + 2i48Ki123- 88cos2e- 68cos4e + 24cos6e
`+9cos80) + ... ,
`
`(12.13)
`
`which includes contributions from all three (K1, K 2 , and K 3 ) terms.,
`Next let us calculate the anisotropy energy in the (111) plane. For this purpose, we
`, y 1
`), in which the Z 1-axis is parallel to the [111]
`set up a new coordinate system (x 1
`, Z 1
`axis, the Y1-axis is in the (1l0) plane, and the X 1-axis parallel to the [1l0] axis (see
`Fig. 12.8). The direction cosines between coordinate axes of the old (x, y, z) and the
`) systems are listed in Table 12.1 *. The (111) plane in the old coordi-
`new (x 1
`, y 1
`, z 1
`(1 1 1)
`l3 , l3 , 13 ,
`* To construct such a table, first we determine two axes, say x' {i , - {i , 0 and z'
`and then by using the relationships I:; a?= 1, I:; f3/ = 1, I:; r? = 1, I:; a; /3; = 0, I:; /3;1'; = 0, I:; I'; a;= 0, we
`can determine other direction cosines.
`
`(1 1)
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`
`MAGNETOCRYSTALLINE ANISOTROPY
`
`z
`
`x'
`
`X
`
`Fig. 12.8. New cubic coordinate system (x', y', z') with x' II [llO] and z' II [111].
`
`nates is the x'y' plane in the new one. Let (} be the angle between the magnetization
`and the x' -axis in the x' -y' plane; then we have
`
`a~= c~s (}}
`a 2 = sm (}
`a;= 0
`Referring to Table 12.1, we have the direction cosines of magnetization in the old
`coordinates
`
`(12.14)
`
`.
`
`1
`1
`1
`1
`1
`a 1 = {i a~ + l6 a; + {3 a; = {i cos (} + l6 sin (}
`1
`1
`1
`1
`1
`l6 a 2
`l6 sm
`'
`'+
`'+
`·e
`a2 = - {i a 1
`{3 a 3 = - {i cos
`{i
`{i
`1
`a' + -
`sin (}
`a' = -
`13
`-
`-
`{32..[33
`
`a =
`3
`
`-
`
`e+
`
`(12.15)
`
`K2
`
`(9sin2
`
`(}- 24sin4
`
`Using (12.15) in (12.5), we have the anisotropy energy in the (111) plane
`K3
`Kt
`(} + 16sin6 e)+ M
`Ea = 4 + 54
`K3
`Kl
`K2
`= 4 + 108 (1- cos6(}) + 16 + ... '
`in which only the K 2 term has angular dependence and contributes to the anisotropy
`energy.
`
`(12.16)
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`PHENOMENOLOGY OF MAGNETOCRYSTALLINE ANISOTROPY 255
`Table 12.1.
`
`X
`
`1
`li
`1
`!6
`1
`f3
`
`y
`1
`-li
`1
`!6
`1
`f3
`
`z
`
`0
`li
`-13
`1
`f3
`
`x'
`
`y'
`
`z'
`
`In summary, the anisotropy energy can be expressed in the general form
`(12.17)
`E. =A 2 cos28+A 4 cos48+A 6 cos68+A 8 cos88+ ···.
`In the case of uniaxial anisotropy, by comparing (12.17) with (12.2), we have
`tKu 2 - ~Ku30+ cos6c,o) + ···}
`A 2 = - tKu 1 -
`A4 = tKu2 + -kKu3(1 + cos6c,o) + ···
`.
`A 6 = - fz-Ku 3(1 + cos6cp) + ···
`In the case of the (001) plane in a cubic crystal, by comparison with (12.11), we have
`
`(12.18)
`
`In the case of the (1l0) plane, we have
`
`In the case of the (111) plane, we have
`
`(12.19)
`
`(12.20)
`
`(12.21)
`
`Anisotropy energy is also produced by magnetostatic energy due to magnetic free
`poles appearing on the outside surface or internal surfaces of an inhomogeneous
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`(12.22)
`
`256
`MAGNETOCRYSTALLINE ANISOTROPY
`magnetic material. Referring to (1.99), the magnetostatic energy due to free poles on
`the outside surface of a magnetic body of volume v magnetized to intensity I along an
`axis with demagnetizing factor N is given by
`1
`U= -NI 2 u.
`21-Lo
`If the specimen is in the form of an ellipsoid of revolution with the long axis parallel
`to the z-axis, the demagnetizing factor parallel to the x- and y-axes is given by
`Nx =NY = 10 - N), where Nz is the demagnetizing factor along the z-axis. Let (} be
`the angle between the magnetization and the z-axis, and q; be the angle between the
`x-axis and the projection of the magnetization onto the x-y plane. Then applying
`(12.22) to the x-, y- and z-components, we have the magnetostatic energy
`1
`U = - - I,2 v( Nx sin2 (} cos2 q; +NY sin2 (} sin2 q; + Nz cos2 (})
`21-Lo
`1
`= - - Is2 v(3Nz - 1) cos2 (}+canst.
`41-Lo
`which depends on the direction of magnetization. This kind of anisotropy is called
`shape magnetic anisotropy.
`If precipitate particles of magnetization I;, different from that of the matrix I,
`U; II I,), have demagnetizing factor ~ (Nz < t ), the magnetostatic energy is given by
`(1,-I,) v(3Nz-l)cos e.
`-
`1
`' 2
`(12.24)
`U--
`4
`1-Lo
`The easy axis of this anisotropy is the z-axis, irrespective of the magnitude of I;
`relative to I,. One example of this kind of shape anisotropy is found in Alnico 5, in
`which elongated precipitates produce the uniaxial anisotropy (see Section 13.3.1).
`
`(12.23)
`
`2
`
`12.2 METHODS FOR MEASURING MAGNETIC ANISOTROPY
`
`The most accurate means for measuring magnetic anisotropy is the torque magne-
`tometer. The principle of this method is as follows: the ferromagnetic specimen, in
`the form of a disk or a sphere, is placed in a reasonably strong magnetic field which
`magnetizes the specimen to saturation. If the easy axis is near the direction of
`magnetization, the magnetic anisotropy tends to rotate the specimen to bring the easy
`axis parallel to the magnetization, thus producing a torque on the specimen. If the
`torque is measured as a function of the angle of rotation of the magnetic field about
`the vertical axis, we can obtain the torque curve, from which we can deduce the
`anisotropy constants.
`There are various types of torque magnetometer.4 Figure 12.9 shows a typical
`automatic torque magnetometer. The specimen S is suspended by a thin metal wire in
`'"
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`METHODS FOR MEASURING MAGNETIC ANISOTROPY
`
`257
`
`Fig. 12.9. Automatic recording torque magnetometer.
`
`the magnetic field between the pole pieces of the electromagnet E, which can be
`rotated about the vertical axis. The lateral displacement of the specimen holder is
`prevented by a frictionless beari