INTERNATIONAL SERIES OF MONOGRAPHS ON PHYSICS

109- A. Aharoni: Introductlon to the J/zdory ofjkrronlnnetigmzle10s. R. Dobbs: Helîum three107. R. Wignans: Calorîmetry106. 1. Kubler: Theor.y ofitbterant electron ZZCJPTIIJILCZZI105. Y Kuzamotoa Y Kitaoka: Znamics ofheapy electtons104. D. Bardin, G. Passarino: The s'ltzrldzprtf.àftltftl/l themaking103. G. C- Branco. L. Lavoura, J. R Silva: CP violation102. T. C. Choy: Effective zneï?.f?n theory101. H.Arnkl-: Mathematical theory t//gaznflwl-/ielé100. L. M. Pismen: Vortices fzl nonlînearhelh99. L. Mcstd: Stellar magnetism98. K. H. Bennemann:aNbn/fnelr optics izz metals97. D. Snlzrnann: Atomicphysicz in hotplnmas96. M-Brambilla:AAetj/ dteory ofplnma waves95. M. Wakatani: Stellarator andhelîotron (fevfce..g94. S. Cbt-kmznrn'- : Physics offerromagnetîsm93. A. Aharoni: Introduct>n to the theo6' offerromagnetism92. l Zinn-lustin: Quanutmfeld t/zeoF.)? andcriticalphenomena91. R. A. Bertlmann:-dnozna/ie.ç /W quantumfddtbeory90. P. K. Gosh: Ion traps89. E. Siml' nek; Inhomogeneous superconductors88. S. L. Adler: Quaternionic quantum mechanics crzdfélgtznr?zra/à/fïç87. R S. Joshi: Globalaapects j;z gravitation tzz?zf cosmolov86. E. R. Pikez S. Sarkar: The gzzm/?zrzl theor.y ofradiation84. M Z. Kresin, H. Morawilas. A. Wolft Merhmn-vnn tl/ctpnpe/ztz'ilzitz/r-z?âfkgâ

Trsuperconductipity83. P. G. de Gennc l Prost: Thephysica ofliquîd crystall82. B. H. Bransden, M. R. C. McDowell; Charge exchangeand the Jâeory of

ion-atom cof/gfon8 1. J. Jensen, A. R. Mackintosh: Rare earth magnetism80. R. Gasuans, T. T- Wu: The ubiquitousphoton79. P. Luchini, H. Motz: Undulators andfkee-electron Jtzvcr,g78. R Weinbergenflclron scatterùq theoy76. H. Aoki, H- Kamimm'a: Thephysies (Kln/dmclfzlg electrons fa disorderedsystems75. j. D. Lawson: Tlzephysica ofchargedparticle beams73.. M. Doi, S. E Edwards: TM Jâeo?:y ofpolymer dynamks71. E. L. Wolf: Princàles ofeleetron tunnelûq spectroscopy70. H-lri-l-lcnischrseznfconlzlcror contacts69. S. Chandrasekhar: The mathematîcal theoty ofblack holes68. G. R. Satchler: Direct nuclear reactiomg51 . C. Mzller: The Jâeor.p ofrelativity46. H. E. Stanley: Introduction tophase rrlrl-Wzïo?z.ç andcrîticalphenomena32. A. Ab ' J'r/rlclèle: ofnurlrmr lmgndl?:î'zzz27. E A. M. Dirac: Princèles ofquantum mechanics23. R. E.peieris: Quantm theoty o-f-çozktç

lntroduction to the Theory ofFerromagnetism

SecondEdition

AMIKAM AHARONIThe Aic/lardfrorl-çfefn Professor ofTheoretical Maznelâg?zz

EmeritusWeizmann lttstitute ofscience

Israel

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1S3N 97$-0-19450809*

PREFACE TO THE FIRST EDITION

This book is mainly intended to be used as a textbook by fzst-year grad-uate students aad advanced seniors in physiœ and en#neering. lt is built,however, in suc,h a way that it em.n also serve as a reference book for pro-fessiolals who work on advanced topics of magnetism, but want to releshtheir previous studies, or look more deeply 1to the basis of what they aredoing. It is based on a course that 1 used to teac.h in the F'einberg GraduateScàool of the Weizmann lnstitute of Science, which hms been widened here,maixly in tûe part denlêng with micromagnetics.

The. emphasis is on explaining the bmsic principles, without going toodeeply into any of thespecial îelds which are normally discussed in diFerentprallelsessions in the magnetism conferences. The idea ks to Tve beginnersas much coverage as is possible within a reasonable size of one book, andto have mature researcàers in one Zeld have at least a glimpse into whatother Eelds are a11 about. Only in the particular feld of micromagneticsdid 1 allow myself to go somewàat into the state-of-the-art of some moreadvanced topics, mainly because there is no comprehemsive treatise whichcovers this sublect, or even any part of it. 1zt some ways, tkis book is nisxnmeact to be suc.h an advanced review of micromagnetics. But even for 'thissubject I tried to concentrate on the b'msis, and avoid most of the technicaldetails whic.h belong in suc,h a treatise. The more advanced parts are usuzly#ven as references to the lkterature) which should help reseircAers v-ithoutconfusing the students.

This book is theoretical, but it is by no means meant to be read ollyby those who wast to become theorists. I have tried to keep in mind a11those practising en#neers and experimental phydcists who only too oflendo good expedmental work, without uderstanding ihe theory behsnd it.They usually look up, and with too high respect, to the theozists whosepapers they are unable to read., mainly because these papers are written inac incomprehensible lMguage. 1 hope they will be able to read thks book,asd to flnd out that in many cases there is nothin.g behind that obscurepresentation, acd the theorists only pretend that they know what they azedoing- For thîs purpose, the emphasis is always on the disadvantages andthe drawbacks of each theory, more than on its advantages, which in mymind are self-eddent. In particular, T keep pointing out the appvowimationswhic,h some theorists ignore, or even. try to Mde, clalmlng that a particularcalculation, or a prticular resui, is ermet.

Tt is mssumed that the reader has already taken an undergraduate courseon âelds, and is fnmêliar with Maxwell's equations, and the way they are

PREPACE

dedved. Some (bat desnitdy not a11) of that subjed matter ks repeatedhere, aad pr-nted 1om a de' erent angle, and whh a more mature point ofq'iew. I iope that this repetition will help establish a better understandingof the ma>etostatics than ks posWble in a typical uadergraduate course,wlkich rushes to cove'r tNe mzrriclzlum, with no time left to understand it.However, even in ths most ilask mavetostatics, I = mostTy tryiag togive the student a good foundation to build the thecay om rather tlun tococer a lot of ground, or to go into tàe Ene detus of particular problems.

M%t of this book usœ chssiG physics oaly, because it Ls Hprwsa-bleto do tt otherwise. In spite of some cblmq by some enthusîasts, thue is noquantum-mehanical theory of magnetsqm which covers more than a minorlittle cozmer of the subject, and even tlïat is done by usîag ver.r roughappremations. I made a spem.- 1 point to discuss tlle Bohr-van Leeuwentheorem, in section 1.3, because it Ls quoted much too often by quantum-mechaniœ spedœl-les, who look down on everybody else, and clpl'm tttattheirs is the only tmle physics. Thox maaetidans who do not developa.a Oeriority complex from tàese aTgttmenis mar sldp the details of thatsection. Nobody ever tells the quantum-mechaaical experts to avoid certainappremations used ia their theozy, and. if told, they could not care less. Iam trying to eatourage the classical-mechaniœ theorists to have the samea:œtude.

Ia the Feberg Graduate School, atl students are rmuired to take a.aâdvanced course in Qaantum Mebanics, on which I could baild my course.TMs cmse, howevez, is aiypical, a'nd ix othe pla- many stadents, mostlyin ensneezlng, tm-acà a graduate school wtthout ever being exposed to aayquantum mehanics. For the beneft of these students, 1 collectu all thequantlzm-meehnm'cal dismeon into chapter 3. Tke rest of the book iswritten witkout anyessential reference to this Gapter, and e.xn be followedevem if nhxpter 3 is omitted. Tkose who How sometkiag 'about quantummechanics shoald be able to beneft from the preenutioa of the lumicprinciples in that Gapter. But those who do not know quantum mechanicscan easGy do 'witkout it.

1 have IZSH aa unonventional order of the subject matter, startingfromexcxaage, tken addingto it the auisotropy, aad induding tlme eects, wlzie.kaze usually studied without any refereace to tàe magnetostatics aayway.The magnetostalk interactions come only after the superparmagndism. Ibelieve that this order is pnqier to follow by students than the order wkichmost of my colleagues wottld have recommended, aad that R àeelps to graspthe prindple beknd the equations.

The referen- are only meaat to indicate where the reuer may îndmore informatiol which is relevant to some points. They are not meaat toheb write the Mstory of the subjed, althougà 1 sometimes mentioa whostrted that line of research. Therefore, the older papeze are not mentionedif they are quoted in newer one, unless the older ones contain (='tG

PREFACB

information which c=not be found in newer ones. 1 never could understandthe point of those who dte the original work of Maxwell, for exnmple, whichnobody bothers to look up anmay. I have also restdcted a1l citations topapers in English only. Some years ago-all students were required to have atleast a workng knowledge of other lpmaages, but these days are passed,and I see no reason to put a list whic,h nobody will even look at. It israther strange to discuss the works of Néel without citing any of his origiaalworks, or even to discuss the Döring mass in section 10-4 with no mentionof Döring. But 1 prefer to do it tkis way, and aaybody wko wapts to readthe papers of Néel in Frenc,h can easily fnd the necessary references ia thepapers whic,h I dte. After all, I xrn not dting all tke older papers in Englisheither, whic,h could easily make a list of many thousands. I was temptedto make a,n Gception izz the case of the Dietze and Thomas model of eqn(8.1.1) wilic,k 1 discuss in much detail) but then I dedded against it.

It may not seem so to those who are used to textbooks with hardlz anyreferences at all, but 1 tlied to keep their number as small as possible, andit is, after a11, also a aide to reseœcllers. Iu as muc,h as po>ible, I nlncj triedto refer to reviews for broader aspects of the topic discussed in the text, orto semi-popular artides i.n Physics Today, whic,h are rathe.r eas'y to be readby beginners- This nzle is not always adhered to, aad in many cases thereis nothing besides the ozi#nal, and diëcult-to-read, aiticles. Occasionally1 may have aISO been carried away, and dted some more advanced articles;wkick are deânitely not on the intended level of the book. But it should beborne izt mind that the beginner is not eoected to read a11 those articles,and some of tlwm are only intended for practising researchers who want togo into some more detail.

I tried to avoid citiz.g conference proceedings which a2e not part of ajournal, and also joulmals wàich are not easily available in many libraries,unless lothing more appropriate is available. For examplea before Brownpublished the full account of what 1 refer to here as (171)1 he published ashort version in 2963 Z Appl. .JW:>., 34, 1319-20- There is nothing in thelatter which cannot be found in (171), and there Ls no reason to mentionit. 'Phen Brown did the s=e with (508), a short version of whic,h wasfrst published as (50z4. But in this case, the h!11 paper is published in anobscure journal, wàick not many have. Therefore, in this case 1 also dtedthe skort version (50:1, so that those who Onnot :nd (5081 caa at leastread something.

&''l!7Wm

Rehqcoih, Jsrc:lDecembe.r 1995

PREFACE TO THE SECOND EDITION

Research nowadays advances vez'y 1st, and only a few years after thisbook had been published, parts of the presentation of micromagnetics hadalreadybecome obsolete, and called for a revision. The major chaages, how-ever, aze mostly conceatrated izl two sections of chapters 9 and 11. The partthat is intended to be used a,s a te-xtbook and Mmost alt the dismzssions ofthe older, and more established, theoretical magnetism remain essentiallyunchanged. Most of the revision in this part consists of correcting 'a fewtypographical errors that 1 have failed to notice even'in preparing the sec-ond prîating of the frst edition, or some minor rephrasing. There are alsoseveral updates and revised discussions of some new references, mostly intopics, such as resonaacesr that were on the border of the scope of the frstedition. '

Of the two major changes in micromagnetics, one addresses the searchfor a Qhird mode'' of nudeation in a perfect prolate spheroid. It was a,nunsolved problem at the time, and its presentation in the prgdous Mitions'zas necessarily cautious and unbinding. The zeader was warned that thepopular search for such a tstllird mcde': was most probably futile, but lhadto admit that there was no proof that it could not efst. The proof waspublished in 1997, and now the presentation in section 9.4 is single-valtted,with a clearer condusion: there is no point ilz looking for such a 'fthirdmode'' because it caanot possibly efst-

The second change deals with the problem of using sharp coMers incomputational micromagnetics. lt îs even more dramatic than the frst one,because this'question has been highly controversial for more than 40 years.lt wms fnally settled last year in a satisfactozy muner, by tke late Ale,xEubert and his collaborators. The new form of section 11.3-5 reêects tllssdrastic c'hange that Ls going to infuence strongly all micromagnetics re-search in the comhg years.

Rehmmthv zàmclJuly 2000 Amium Aharoni

CONTENTS

1 Iutroductiong:t.:L Nomenclaturel,2 Weiss Domains1.3 The Bohr-van Leeuwen Theoremi, Diamagnesism.4

2 Molecular Field Approxîmation2.1 Paramagnetismk.2 Ferromagnetism2.3 Antiferromagnetism2.4 The Curie-Weiss Law2.5 Ferrimagnetism2.6 Other Cases

3 The Heisenberg Hnrniltonian3.1 Spin and Orbit3.2 Exchange Interaction3.3 Fuxckanje lhtegrals3.4 Delocalzzed Electrons3.5 Spin Waves

4 Magnetlzatlon vs. Temperature4.1 Magnetic Domnsns4.2 The Landau Theory4.3 Critical Exponents4.4 Ising Model4.5 Low Dimensionality4.6 Arrott Plots

5 Anisotropy aud Time EFects5.1 Anisotropy

5.1.1 Uniafal Anisotropy5.1.2 Cubic Anisotropy5.1.3 Magnetostriction5.1.4 Otker Cases5.1.5 Surface Anisotropy,

. 5.1.6 Experîmental Methods5.2 Superparamagnetism5.3 Magnetic Vkseosity

11368

12121620222732

35353643

48

60606366707680

8à8385868788899092

100

x CONTBNTS

TLe Stoner-s7ohlfarth lfodel

6 Another Ene'rr Teerm6.1 Basic Magnetostatics

6.1.1 Uniqueness6.1.2 Triviaz Ex=ples6.1.3 Uniformly MagnetizW Ellipsoid

6.2 0rigi.a of Domains6.2.1 Domain Wall6.2.2 Long and Short Range

6.3 Magnetic Charge6.3.1 Geae'ral DemMnetizatioa

6.4 T/zdts

7 Basic Micromngnetics7.1 Iclassical' Kxchaage7.2 The Landau and Izifshitz Wall7.3 Magnetostatic Energy

7.3.1. Physieally Small Sphere7.3.1 Pole Avoidaace Prindple

'

7.3.3 RHprocity7.3.4 Upper and Lower Bounds7.3.5 Planar Rectrgle

8 Energy Mlnlml'zation8.1 Bloch and Néel Walls8.2 Two-dimensional Walls

8.2.1 Bulk Matfm-nqK8.3 Brown's Static Equations8.4 Self-consistency8.5 The Dyn=ic Equation

9 The Nucleation Problem9.1 Deinition9.2 Two Eigenmodes

9.2.1 Coherent Rotadon. 9.2.2 Maretization Curling

9.3 Infnite Slab9.4 The Third Mode9-5 Brown's Paradox

9.5.1 Eazd Materials9.5.2 Soft Materials9.5.3 Small Particles

3.0 Analytic Mkromagnetics

705

109109I1l112114116120122125128131

133133138141142145148149152

157157165171173179181

183183188188189194200204207209212

215

CONTENTS X4

215215217221222225226231232

238238243250250252255256260266

268

315

10 Analykic Micromagnetics10.1 F erromagnetic Raonance10.2 First Jntegral10.3 Voundary Conditions10.4 WaII Mass10.5 The Remauent State

10.5.1 Sphere10.5.2 Prolate Spheroid10.5.3 Cube

11 Numerical MictomagneticsA1.1 Magnetostatic Enera11 2 Energy Minl'mization11.3 Comptltational Results

11.3.1 Domain Walls11.3.2 Sphere11.3.3 Prolate Spheroid11.3-4 Tln-n Films11.3.5 Prism11.3.6 Cylinder

References

Author Iudex

Subject Index

CONTENTS

INTRODUCTION

)1t $1 NomenclatureIt 'id Mown from experiment that every material which is put in a magneticfeld, H, acquizes a maretic moment. The dipole moment per unit volume11b defned as the magnetization, and vdll be denoted here by the vector M-V most materials M is proportional to the applied freld H. The relation' )ù then written as

M = xH,

and x ks called the magnetic awuscegtfàdlïtp of the material.Maxw-ell's equations are usually wrîtten for the vector

B = Jzc (H + %M),

instead of M. Here rtn is a notation introduced by Brown I1) to includedlfferent systems of units. ln pariicular, % = 1 for the S1 uztits, whîch aa.epopular in textbooksr while % = 47 for tite Gaussian, cgs units, which are.most popular among magneticians: and for which p,o = 1. If ecn (1.1.1) is'Vllilled, it is also possible to revite eqn (1.1.2) as

B = ;&n,

wherep = p*(1 + '/BX) = Jttp,r (1.1.4)

ib knoqqz as the magnetic germecàïlït:?. The material is clx--fled ms Cpararmagnedc' if x > 0, f.6. ys > 1, and as 4diamagnetic' if x < 0, i.e. p. < 1.

There are, howmrerj some mateziats which do not ft this classif.cation,because in these materials the mawetization M is not proporkonal to tiexpplied feld, H. lt may be, for exmple, non-zero at H = 0. Actually, L6,1 injhese materials is not even a one-valued fmnction of H, and its value dependson the history of the applied ûeld. A typical case is shown in Fig. 1.1, whichplots $he component of M in the direction of the applied feld, MJJ, as a'function of the magnitude of that feld- The outermost loop is known as thelimiting àpstereyf.s c'u'rt)c, and is obtained by applying a suEciently Iargedeld in one direction, decreasing it to zero, and then incremsing it to a largevalue in the opposite direction. The curve is reproducible in .consecutivecycles of the applied îeld.

2 DITRODUCTION

alplf';rdr

S

T

/z'

H Hl tt

I

iI

FIG- 1-1. SGematic representation of the limiting hysteresis curve (orloop) of a typical ferromagnetîc matezial, displaying also the virgincuz've (dashed), acd one minor loop. Also shown are the remanence,Mrt the saturation maaetization, Ms? and the coercivity, Hc.

The curve which starts at the origin is known as the uivgin znc-kn6tïzctioncurne, and e-an only be traced once aler demagnetizing the swple, nxmelybringing it to a state izt which M = 0 at H = 0. This demagnetization maybe acMeved by heating the snmple to a high temperature, and coolingit in zero Eeldj or by cycling the applied feld with steadily decrea-singamplitudes. Iï the feld is increased, then decwreased before the limitinghysteresis cuz've is reaGed, ar.d then the îeld is reversed, a so-called rrànorhysteresis Je@J Ls traced. One exaaple of mzc,b a cuzve is shown in Fig.1.1, but there is actually a whole continuum of them. With an appropriatehistory of the applied feld, one can therdore end at aay point inside theltmiting hysteresis loop. J.n particular, it is possible to reach H = 0 withany p-alue of MH betweea -l% and mMr, where Mr i.s the value of MHon tke ll-msting hysteresis cuzwe, at H = O (see Fig. 1.1). It is called theremanômce or the remanent rncgnetfzctïtm.

It is possible) although not rtuxy necessary, to defne pfarrne-ability forferromagnetic materials, in order to pretend that they are similaz in some

xay to panmagnetic mateaiis-ther eqn (1.1.1) nor eqn (1-1-2) is G'l6lled in fenomagnetic materials. Nev-eohezless, it is quite mlstomary to introduce some 6.Tctin6 permeability ata particular wlue of the applied Eeld, H, as

'Wrslss DOMATNS

Jzqff = OBHI8II,

3

(1.$.5)

or, over a certain range of Nalues of Ih a,s

ltes = LBHILII. (1.1.6)J.u either case: it is possible to a certain approfmation to apply Maxwell'sequations to ferromagnetic materials in the same way they are nsed forparamagnetic materials, with tids eFective permeabzity- However, here M isnot a constant ai it is in paramagnetic materiaks and the whole formulation)is at best usable at a particular applied feld.

Two other important terms are azso defned on the limiting hysterciscurve in Fk. 1.1. One is the coerci6sy or coertbt)e jbrce, Hct whic,h Ls thevalue of H for whic,h Ms = 0: and is actually a magnetlc fe-ld aad not aforce- The other is the satnration mcpnetïzcfïon, or spontane@'as t'ncgneàï-zationt Ms, whick will be deined for the meantime as the value of Ms , orthe magnitude of M, ja a vezy large feld. This defnition is not accurate,and w'ill be modifed in section 4.1, but it should do for now-

Thks satm-ation magnetization is an htrinslc property of the material,and is independent of the sample, lf properly measured. It is a fundion oftemperature, a typical form of which is plotted in Fig. 1.2a In this fgure,Ms is normczed with respect to its value at zero temperature. The tem-perature is normalized with rcpect to the so-called C'urïe temperatnret Tc,of the materlal, whic.h is the temperatare at whir,h Ms becomes 0 tzf zerocmlbed $eld. When nozmalized in this wqsy, the cun'es for diferent ferro-magmets aze very nearly the same ms in Fig- 1.2. .X11 ferromaaets becomeregular pazamagnets at temperatures above Tc; and as such they have anon-zero magnetization in the presence of the feld which is used for themea-surcent. This beaviour is emphasized in Fig. 1.2 which shows thecurve as adually measured in a small applied feld, at which 2)G does notgo to zero at the Curie temperatnze- Speec exmerimental mlrves for Niand Fe cxn be seen, for ava.m ple, in Fig. 9 of Potter r2).1.2 Weiss DomninsIn principle, any theozy of ferromavetism should address both of thase un-usual phenomena, which are not encountered in other materials. It shouldthus erplain the hysteresis displpyed in Fig. 1.1 qnd the temperature-dependence of Fig. 1.2, even though most theoreticians work only on oneand iaore the other. They do it even when they compare the results withazt w'tperiment) which almays involves both. It is inevitable, because thegeneral Luuantitatine problem is too complicated for the present state ofHowledge.

Qualitatively, both phenomena are understood to a certxin extent dueto a,a explanation already given by Weiss in 1907. Weiss asumed that there

WTRODUCTION

1

0.8

7

o'x'.s 0.6>NxD I=tA 0.4 =

0.2

l I . 1 I I --u- . ' .' I0'

0.2 0.4 0.6 0.B 1

nG. 1.2. SpontanYus magnetization, Ms, of a ferromaaet as a functiouof the temperature, T, normnlized to tlke Curie temperature, Tc. Theapplied.feld is Msumed to be small, but 5:111,e, ms it is in real m-nxm.e-ments.

Ls a certain internal (or Cmolnruxlar') Eeld in Terromagnetic materials, whichtrie to align the maretic Gpoles of the atoms agahst thermal fuctuationswhicà prefer a œmplete disorder of these dipolœ. As will be sœn ia thenext chapter, such a molecularseld is suRdent to explain the temperaturodependence as plotted in Fig. 1.2, and the paramagnetisan ibove the Curietemperature. However, 1.hz.-R model leads to a cdntdczàà maoetization Mat any sven temperature below Q. ln orde to explain the unusual feld-dependence in Fig. 1.1, Weiss Msumed that ferromagnets are madè out ofm=y domains. Esmb of these domxsns is magnetized to the saturation valueMs(T) as i!z Fig. 1.2, but the diration of lhe magnetization vector vazies

'

1om one domaiu to the other. The measured value of the magnetizationis the average over thex domains, whic,h catl be zero in any particue

WEISS DOMAINS 5

diredion when there is an equal number of domains pazallel and antiparallelto that diredion. It ca.c also have a, non-zero value, numericazy less thaztMs(T), if this number of domins is not equal. The applied magnetic feldrotates the ma>etization of the indi'ddtzal domains into its own direction,and when this feld is sulciently large to align a1l domains, the measuredaverage value beœmes Ms(T)- Without going into fne details, it should bequite clear that this assumption is suëcient to eolain the Geld-dependencein Fig. 1.1, at least qualitatively. '' .

Weiss did not justify any of his two azbitrar.g %sumptions, and couldnot explain the origin of the molecula,r field or the efstence of the domains.There were also several diëculti% in implementing his principles for anyquantitative estimations. In pazticulasr, using the experlmental value of theCurie point in such a theozy, the magnitude of the molccular feld i.tl kronturas out to be of the order of 106 Oe. lt takes a êeld of the order of 1 Oeto rearrange the domains in ixon, and 103 Oe to elp'rnlnn.te them altogether.How come that a feld of 106 Oe is not suscient to align clf the magneticmoments of iron, arzd it takes a.u extra feld of only 103 Oe to do ît? Andhow is it that. even a 1 Oe îeld can contribute so very sfgnl-ficantly towardsa task whic,h a 106 Oe fcld Ls not suëdent to accomplish?

In spite of these disculties (which v'itl be addressed later) the assump-tions of Weiss aze actually valid aad sound, and contain the basic under-standing of ferromagnetism. The molecular feld is lmown now to be acertain approfmation to a coupling force between spins, which can be de-rived âom more basic prindples, as will be shown here at diferent levels.The Hstence of domains, magnetizcd in dferent diredions, is not-even anassumption any more. Th%e dömains have been obsered by several tech-niques, outzined ia section 4.1, and their Hstence is now an establishedeoerimental lbct. The only diference is that they aze lmown now to bemagnetized along certain directions; and are not a-s rnndomly oriented asWeisfthought. However, thdr Gstence being an experimental fact shouldnot stop us fzom entuizing 'tnlq these domains e-xist, and an appreciablepart of this book q'i.1l be dcvoted to answerkng this question, a-s well as thequestion of why a 106 Oe îeld cannot do what a 1 Oe 6e1d can.

Even though the basic properties of ferromagnets are quantum mechan-ical by nature: most of the treatment here will use dassical physics) at' alevel whick can be followed by engineering students who did not take anyquantum mechaakcs in college. lt is not only the choice for this book. Mostof the development of the theoz'y of ferromagnetism was done using classicalconcepts only, even in recent years when evezybody knows that a classicaltheory can at best be only aa approfmation to the true quantum treat-ment, enecially in magnetism. The reason is that pure quantum theozyhas not Mvanced yet beyond simple casu which are of very little practtcalapplication. However, before adopting tMs classical approach to ferromapnetism, it is necessazy to consider a famous and ofken-quoted theorem of

6 WTRODUCTION

Bohr and van. Leeuwen, accordhg to which cl%sical physic cannot possi-bly lead to magnetism, because in pure clnMical physics the electrons in amat-rlltl do not interact with a.n applied magnetic ield.

1.a The Bohr-vazz Leeuw-en TheoremConsider a claôsicai syste,m of N electrons. They have 3.6r degree of 1e0.dom, and are therefore descibed by their 3wV coordinates, qq, and their3N momenta, pé. Eac,h electron has a (negative) chargej d = -!:g1- Iu cgsunits) aa electron whose velodty is v creates a curzent densityj = cv, anda magnetic moment

. (t em = ygr x J = sr x v, (1.3.7)

aat the position r in space, where c is the velodty of light. The importaatfeatureis that this m is a Iinqar fanction of the velocity v of each pazticularelectron. lt mpltns that whatever the patteM of motion of a11 the eledronsks, the total magnetk moment is also a linecr function of all the electronvelodties- Therefore, the a-componeat of the total magnetic pomeat of allthe electrons must be a funaion of the form

3NJ (çz , - - - , q'nr ) q-i ,T)l> = GL

2=1/1.3.8)k

where the dot designates a derivative with respect to timej and where thecoeëdents, a(, are functions of all the coordinates %, but do not dependOn R .

The canonical etpations of a classical motion aze

J>f &/fié = , 3% = - (1.3.9)opi t'kï

where

(1.3.10)

is the Hnm-lltonian, m. is the electron massj A is the vector potential ofthe magnetic feld, and c7 is the potential eneroe. Substituting eqn (1.3.9)

'

in e1n (1.3.8)j3N

'rn,.'s = tz'; (ç17 - - - , :3.N') a - (1-3-11)Pif=z

If k'B is Boltxann's constant, T is the temperaturez and

1/3 = ,*T (1.3.12)

TI'IE BOM-VAN LEELTWEN TROREM

the dassical statistical average whic'h will be measured Ls

) pue-Rdqz . . .dqsxdps . . . dpzxMz = .

J e-sdqï . . . dqsxdpï . . . dpsx (1.3-13)

According to eqn (1.3-11): the numerator in eqn (1.3.13) is a sum ofterms, each of which is proportional to

O #7Y-rté =-c Tit'?py-<Z C

* WY 1 X( - J-sv j m - xP (1.3-14)

whic,h 'kanishes, because H is proportional to W for large tpgi accordingto eqn (1.3.10). Therefore, Mz = 0 and there is no ma>etic moment atany vector potential (namely for any applied âeld), no matter what theactual motion of the electrons in the material 1. i.n other words, there isno interaction between an applied magnetic feld and the eledrons in anymaterial, ifthese eledrons behave according to thelaws of classical physics.lt meaus that dassical physics cannot account for either diamagnetism orparamagnetism, 1et alone ferromagxetism.

This theorem is ver.v general, aad its proof is rigorous. Eowever) it do%nöt eliminate all possibility of usiag classica,l physics. A11' it elirnl-nates isthe use of pure classical physics, which nobody is doing aayway nowadays.Classical eledrons cannot move in a cirOlar orbit azound the atomic nu-cleus without radiating their energy and collapsing into the centre. Butm=y of today's ûclassical' theories only use the result of qurtum mechan-ics to force the electron into such orbits) and calculate its radius classically.Izt fact this circular orbit is a11 it takes to allow a clmssical theory of diamag-netism, as will be seen in the next sedioa. Classical eledrons do not havea spin either. Bat by superimposing the quantum-mechanical concept ofa spin, a classical theory of ferromagnetism becomes possibie. Actually, inthe case of ferromagnetism it is not even a real classicaltheory, like the onewhich serves as a limit to quautum mechanics in other felds of physics. Itis a quasi-classical approach which takes the quantum-meeanical conceptof a spin and treats it c.s %? it 'tller6 c classjcak 'pcctnr. lt essentially usesonly a classical Jorm to dress up some quantum-mechanical results, whic,hat frst sight does not even seem aesthetic.

In principle it would be nicer to treat the whole feld of magnetism ingeneral, aad ferromaaetism in particulam, by pure quantum mechanics.Some books, e.g- Wagner (3), and many of tke research papers whic.h arepubished every year: adopt this approach.Eowever, quaatum mech=ics isonly applicable in a vep- limited pazt of ferromagnetism. For G the otherproblems in this Iield, there is just no other choice. They can be eitherignored or studied by the quasi-clusical techniques, as used i!l most ofthis book. Moreover, classical calculations can give a usdul intaitive guid-

8

aace which is suRcient to pzefer them even in some of the cases for whichquantum calculations are possible but complicated- Besides, mauy of thereported quaatum calculations use rather rough approfmations, as will beseen 5n chapter 3. lt has never been established that these approfmatioasaze any better th= the use of classical physics.

WTRODUCTION

1.4 DinmagnetismAs a,n illustration, diamagnetism will be studied here both 1om a qu=tum-mechanical staadpoint, and 1om a quasi-classical approach on which theBohr orbit of an electron is superimposed.

Consider frst an isolated, quaatum-mechanical atom which ha.s Z elec-trons. Its Hnrnlltonian Ls

z a1 e7t = ame y.Y!, gpï - -cA(rç)1 + other terms, (1.4.1 5)

where the other tezms include interactions between the electrons aad thenucleus, and between one electron and another, which do not play a role inthe point under discussion. If the applied magnetic feld Ls parallel to thez-axis) and is constant in both space and time, the vector potentii is

HA = - (-p, z, 0) ,2 (1.4-16)and

H Hpt - A(n). = .--lzpv - ppzlï = -a ndpi s (1.4-17)!.!

where f, is the z-component of the orbital angulaz momentum.Substituting ecns (1.4.16) an.d (1.4.17) in eqn (1.4.15), the expectation

value of the Harnlltoniu isZ z z1 w eHlt e JJV = X(2 ps - 117 + zj a (z2 + &2)f + other terms, (1.4.18)2z?u c cï=1

which leads to a magaetic moment

Z('?V E eH .

iwv- = - = )-) & F7zo - (z2 + p2 ) y .

DH zzrsec 2cï=1

For a mole of the material, the susceptibility ts

(1.4.1 9)

2 Z:J.Z e x;x =

o.a = -

nmea E Cz2 + &2)ï ,f=1 .

(1.4.2 0).

where N ts Avogadro's number, namely the number of atoms in a mole.

DIAMAGNETJSM

In order to obtain the same result by a quasi-dassical estimation, con-sidtr aa dedron modng ai a ccmstant frequenc.y fzc ia a drrnln:r orbit witha radius r, around a nuclems wkose electzi.c charge is Srei. In eq'lilibriumthe œntrifugal force on tbe eledron is equal to its Coulomb attrRtion tothe nucleus, namely

2n'tevz'r = -2r

tdc = .3TIVT

(1.4.21)Hence

(1.4.22)

If a maoetic field, H, is applied, a Lorentz forcej (e/c) v x H, is added tcthe predous forca, the frequeuc'y changes to (tn aztd the new equilibriumequation becomes

Ze2 lelYrSmcto1zz = (1.4.23).2 c

Using the notadonIelealz = , (1.4.24)2mec

eqn (1.4-23) becomeY2 - 2=:7:, - oJ:2 = 0. (1.4.25)

Since LOL <<t a;c even for the largest H whkh e.xn be tecbnically achieved,the solution of the quuratic equation (1.4.25) may be written as

(1.4.26)

The appli-tâon of tbe magnetic îeld has, tkus, shled the fzequency bythe amount ulzt which is called tbe Lavmor Jreçuencp.

This c'hanze in frequency mpmns tkat the electron makes extra *z,/(2x)revolutions per second, tkus creating the additional cmq'ent

(1.4.27)

In cgs units, the maaetic moment Ls j/c multiplied by the area unde.r theorblt, lmmely

2 + .2 gs (az + (ac.a7 = xz + ogo o

J = - .2*

(1.4.28)

which lends lo tbe .:a,z,,e susceptibility as in eqn (1.4.20).nl's result does not contradict the general Bohr-ma Leeuwen theorem,

because a true clasdcal electron c'aztnot sneztlm the orbit assumed in tbis

10

tclasicalh calculation. It does show, however, that correct results may alryobe obtained by this quasi-claasical approach when it is combined qdth someartifacts fzom quantum mecbanics, suc,h as the electron maintnsning thecircular orbit in this case. Ia a sim:-lnm way, the quœsimclassical study offerromagnetism is obtained by hcluding the electron spiu, with some formfor the e-xchange interaction between these spins.

Before concluding the present cmse, several comments are in order. Hthe frst place, the picture of an electron completely localized at the atomicsite is oversimplled for most solids, and it certinly do% not apply tometals, in wkich the degenerate Fermi gas of the electrons occupies certainenergy bands that are split by the magaetic fdd into the so-called Landaulevels. lt is a completely dferent problem, which is beyond the scope of thisbook. The calculation as presented here actually applies only to noble gases,or other gmses when they are ionized down to the complete shellj.. Jn thtte

the material is completely isotropicj and therefore V = y2 = Xr2.cmses sFor thœe cases) eqn (1.4.20) is usually found in the literature as

INTRODUCTION

z r Ze A --x -a-y: = -

gzyz rz 1 - Tç ?r''-' (,= 1

(1-4.29)

and tke T%t of the calculation is the evaluation of Fpl which ks rather simple2for a hydrogen atom, but less so for other matedals-

The second point is that the diamagnetism (ie. negative susceptibmity) of eqn (1.4.20) or (1.4-29) efsts in cJJ matedals? including paramag-nets. Aowever, in paramagnetîc materials the positive susceptiblty is muchlarger than the negative part, whîch is negligible in comparison. in cgs 'Imits,a typical.value for the &st term i'n the square brackets of eqn r1.4.19): AEJ,is of the order of 10-27 while the second term, lctlfzz + v2/(2c)) is of)the order of IO-ZGSL The largest feld H whicb is physical!y attainable is106 Oe, and even at that feld the second term of eqn (1.4.19) is aegligiblecompared with the frst one. The only case when this second term (namelythe diamagnetism) is measurable is when the frst term vaaishes, whicbusually means a closed electronic shell, as i.!l the noble gases. It should benoted that a zero contribution of the orbitql angular momentum, as ford-electrons, is not suRdent, because the electron spin contribution to theîrst teerm would normal!y make the second term negligible annray . Thesecond term is measurable only when the total contribution of orbit andspin to the frst term vanishes, and the single atom does not have any mag-netc momeut in zero applied feld. This is the c>e of diamagnetism, whenthe magnetic moment of eack atom, and of the ensemble of atoms in thematerial, is proportional to the applied magnetic feld, and is always in theopposite direction to that of the feld.

Paramagnetic materials have a non-zero magnetic moment on eac.h of

DLWAGNETISM 11

thdr atoms, whic,h Ls not caused by, and is independent of, azk applied mag-netic feld. A magnetic âeld arranges these moments izz its own direction,so that tke moment increases with an increasing magnitude of the feld,and the susceptibility is positive. The diamagnetic, negative susceptibilityezsts in these materials as well, only it is usually negligible compared withthe positive susceptibility part. In the absence of an applied 'feld, the =n.g-netic moments of the iadividual atoms are randomly oriented, so that theirctlcmçe Ls zero.

Iu ferromagnetic matezials the magnetic moments of the individualtoms interact strongly with e2.c.11 other. As will be seen in the 'follow-a

ilzg, this interaction creates a certain devee of order even ilz the absenceof an applied âeld- This order is the cause of a non-zero average magneticmoment in zero feld, whicll is the basic dfereace between paramagnetismand ferromagnetism. Thus, the classifcation of materials can be expressedia terms of the magnetkc moment in zero applied feld. In diamagnetic marterials, this moment is zero for each atom; in paramagnetic materials tMsmoment is non-zero for eack atom, but averages to ze'ro over m=y atoms;and in fezromaaets even the average is not zero.

2

MOLECULAR FIBLD APPROXIMATION

2.1 Parnmagnetksmlt is necessary to undezstand parn.magnetism before trying to understandfezromagnetism, and it is best to start 1om the same qumsi-clmssica.l ap-proach whïch will bc useful for ferromagnetism. A quantum-mechanicaltheory of paramagnetism E3j is not particularly diEcttlt to htroduce, but itwi'll not serve this purpose ms efecdvely. Therefore, we start by considerîngan ensemble of atoms, and mssume that each of them has a fxed magnedcmoment m. In order to be used later for ferromagnets, we just adopt thequantum-mechn.nscxl result that the magnitude m of this moment m isggBS, where g is the so-called tlaandé fador' or lspectroscopic splittinghctor' and

, jsj:P,B =2rzcc

is known ms the Bohr magneton. lt is another way of expressiug cxqn (1.4-19)1although this ,magnitude does not even need any justlcation in the presentconte:t of paramagnetism. lt may just be taken as a devnition of the atomicmoment. Eowever, there is another quantum-mechanical property of thespin number, St which is convenient to adopt without considerîng the de-tails-lt is that the component Sz can only mssume the 2S+ ldiscrete values-S, -S + 1, - - ., S, ï.e. one of the integral or half integral sm2ue.s between-S and +S, in integral steps.

These magnetic moments are assumed to interact wîth an applied mag-nedc ield, H, but not to interact with each other- The ener> of interactionof a dipole moment m with the ield H is lœoqm to be -m - H. J.f the di-recEon of H is chosen as the z-n.'r1q, the average component of m in thatdirection at a temperatnzre T is, therefore, accor&ng to the classical statis-tics

Zm,emMfLf(O%J) == *

Temxnll ' (2.1.2)

where p is dezned in eqn (1-3-12)- Using the above-mentioned mxgaitudeof m in mzt and noting the allowed values of Sz,

PARAMAGNETISM 13

with,: S

D = S e##R7Sn = S C1rt=-S =-S

(2.1.4)

where tNe notationgn,ïf

,r; = - --..-, (z.1.5)NTks used for short.

The sum in the denominator is the well-known geemetric Kzies

.%' 1 1 1 - f2d+' f-S - ïS'+l5-2 C = -'y (1+4+. ' '+dS) = .-g.- = - (2-1.6).$ ï 1 - ( 1 - (n=-S

Diserentiating this eqttadon with rœpect 1z> (' an.d rearran#ngvJ .s'+1 - s-z q - fsv ..y ..:% jp ts . j-s hS(( - 'i' ; k 2 qhy . J, -1 = .

( 1 - ( h) 2a=-S '

Substituting eqns (2.1.6) amd (2.1.7) in eqn (2.1.3),(zlzz) S((S+: - j-S-1) - LS + 1)(fS - j-S)qsxi = = .t -s - j,s+-z)g;tn (: - 1)(ïzssinh ((S + 1).4) - 2(S + 1) sinhtsz?)-

s+z+sss-, - (fs+f-s) , 4.1.8)fwhere z) is defne in eqn (2.1.5). Therdore,

S (sinh ((,s' + 1)zj - sinhtspl) - sinhtsz?)(Szi = s )

-cosh E(5' + llq) - cosht T? (2.1.9)

Tilis eeression can be simplled by using $he foEowing, well-lmown relartioms Mweem the hyperbolic fcnctions:

#,) cosh (2S -f- lyj , (z.1.1(j;sin.h ((J + 1)q) - sinhtlv?) = 2 sizh (z )

!. funll 25 '6- 1cosk ((S + 1/3) - coshtsq) = 2sinh (z ) !y n . (2.1.11)Also, the last tarm in the numerator of eqn (2.1.9) is transsormed accordiagto

25 + 1 q.sinhtsz?) = sinh r? - =2 2

14 MOLECULAR FTRT.D APPROMMATION

25 + 1 !! 2.9 + 1 p.= sinh z T cosh (z) - cosh z T sin.h (z) . (2.1.12)

Substituting all these in eqn (2.1.9): and dividing by S,

(2.1.13)

Recalling the defaition of 'c in eqn (2-1.5), this relation c.an be written inthe form

( S z ' ) IET--..S S

- . = ss , (2.1.14)S WTwhere the A:nction

25 + 1 2S + 7 1 zSa@) = as

COth g-s- z -

g,j coth (z,j,) (2.1.15)

is cluled the Britlonin Azlcjitm. Besidès the argument z it also depends onthe spin numbert S- .

As an illustration we shall xlqn consider separately the pmicalar caseS = 1. The allowed values of Sz in this case are zg = +la , ard the summa-Rtions ln eqns (2.1.3) and (2.1.4) are over these two values. Therefore,

(&) 28 + 1 28 + 1 1 ?l.s

= -

a s c0th y zz - g c0th ( z ) .

n.n .1. '-nla - ..7/2Ewns -ae + te 1. .; :(Sz.b '.'.w. .u -e,zv = '-

eznfl o en,.'g = -2 tanh (j . :2-1.16)a

That means.

ggp sjy(5'a/ = S tanh -

t (2.1-17)YTinstead of eqn (2.1.14). Thereis no mistake in this akebra) and actually theresults are the same. R can be proved by using the appropriate relatioasbetween the hyperbolic functioms that the de6nstion in eqn (2-1.15) indeedleads to

B Jx) = ta.nh :c. (2.1.18).i kHdwever, this spedal function does make the case S = i) rather atypical.The reader is thus wnrned that a phrase like 'consider for example S = .1.:2is encountered very often in the literature; but more often than not it rdersto a special case, which is dl'eerent 1om what happens for any other wulueof S, even though some salts A with S = .) do ezst.

For smallN-alues of the argument,

1 z sc0th z = - + - + O(z ).z 3 (2.1.19)

PARAMAGNEKSM 15

Substituting in eqn (2.1.15))

(2S + 1)2 - 1 a S + 1 sSs(z) = z + O(z ) = z + 0(z ). (2.1.20)1252 3,5

Therefore, if H is not too large, eqn (2..1.14) becomes

#Jzp,S(.5' + 1)(Sz) = jsz H ) (2.1.21)

which is of the form of eqn (1.1.1). For most parnmagnets at ordinanrtemperatures, no deviations from the line.ar behaviour of eqn (2.1.21) e-an

be detected even for the largest possible feldj H. In the cases for whichthe magnetization is not proportionat to the Geld at high applied felds, itis customary to conserve the fotm of cqn (1.1.1) atorvayp but defne a Geld-dependent susceptibilits x(Jf). The value in eqn (2.1-21) is then referredto as the initial s'tzscepàflflfsp

DLM' a ) an%rgiœ (&) CXrnîtuz = lim - = 1% = -a

H,--& t').s H...n Z'9JJ r (2.1.22)

where N is tke nMraber of spkns per unit volume? aud

h-S L'S + lh- - . . J. 2c = (pJzs) -3ks (2.1.23)

The temperature-dependence of the susceptibility i.!l eqn (2.4.22) agreesM'ith experiment for a11 paramagnetsp ard is lmowu as the C'urie Jcv)- Jkmay aISO be worth mentioning thatr although x may depend on S in para-magnets, it does not depend on the history of H, as is the case for theferromagnets which will be discussed later. In paramagnets x(S) Ls a well-defned, singlo-valued function.

For a very large argumentr cothz -+ 1, and eqn (2.1-15) implies

#s(+=) = +1. (2.1.24)

It means that at ve,zy large applied felds the magnetization saturases aaddoes not keep increasing with the ield. This saturation obviously occurswhen alt the spins are aligned in the direction of the appiied îeld, becausein these parnmagnets the e:ed of the feld ks only to change the directionof the individual, fxed magnetic moments. It does not chrge their magni-tudes, except for the small, djamagnetic contribution, mentioned in section1-4, which always ex-istsj but is usually neglcbly small in parnmagnets.

This saturation caanot be observed in most pnmmagnets, because theavallable feld H is notlarge enough to reach that region-lt is seen, however)

16

that the argument in eqn (2.1.14) is SHIT, rather thaa just H. Therefore,tMs sattzration can be attained at very lOw temperatures; when T is small.It cxn also be observed if S Ls large, which cmm be achievcd by a specia!phenomenon known as sktpenmramannetésm. In normal materials, S is thespin number of a single atom, and is of the order of 1. However, undercertain conditions, which will be descrîbed in section 5.2, S îs the resultantof the spins of many atoms which are coupled together. In these cases Scan be of the order of 103 or 104) and rather small H is snlcient to reac.ksaturation.

MOLBCULAR FDLD APPROXIMATION

2.2 TexromagnetismUnlike the paramagnetic atoms of the previous section, which interact onlywith an external magnetic feld, the atomic spins in ferromagnetic materialsinteract with each other, each of them tr/ng to align the others in itsown direction. Thisinteraction between them originates from the quantum-mechanîcaâ properties of spias, whkh will be discassed in the next chapter.Readers who do not kmow any quantum mechaaics may sldp that chapter,which is not essential for following the rest of this book. They may justadopt as an aMom the efstexce of such a force, which tries to align spinsby the so-called ezchange interaction. The latter can be expressed as anezcltange e'nerp:y between spin Sf artd spin Sy, which is proportional toSï - Sy.

Including the same enera of interaction with an applied Eeld, H, as inthe cmse of paramkgnetic atoms, the total energy of a system is thus

* /E - -5! 74ysç - s.ï - E ggnsi . Rp (2.2.25)

ç/ ç

Khere the prime over the Erst sup indkates that the case ï = J Ls excluded,because the spins do not interact with themselves. Except for these mlues,b0th summations extend over all the atomic spins in the material. Thecoeëcients Jii are called the ezcàtmpe fnteprcls, aad can be eeaated bymethods dœcribed in the next chapter. It should be noted that the sign ofthese coeRcients is defned so that if Jy is positivej Nrallel spins have alower enera than antiparallel ones, whzch is the cmse for a ferromagneticinteractîon.

Vezy many bodies are involved, and some approfmation ts inevitable.In this chapter we introduce a populaz technique, which is Hown by othernames h other branches of physics. In the context of ferromagnetism itis lmown by the name mnlecvlcr neld tpgrozfzrùcàïo'n, although more re-cently the name mean Jcld approzimation is becoming more widely used.In this method, one spin is tagged for checldng its statistics in more detail,while the others are just replaced by their me= value (or, rather, theirquantum-mechanical expectation value). Then, after some maztipulations,

FOOMAGNETISM

that partîcular spin is 'untagged, saying that on the average it is not anydiferent from the other spins, thus obtining the mean value.

Spedqcally, we consider the spin S: ms something special, and collecttogether the enera terms in which it is hvolved (or the I-lamiltonlan whiclzacts on S:), when KE the other spins, Sj, are replaced by their mean value,(S#),

Ei = -2 Jusâ - (%) - J/ZBSC ' H = -Sf ' Hf,#

(2.2.26)

whereH: = p/zsn + 2 Ju(S,) .

i(2.2.2 7)

The factor 2 comu fzom the fact that the double sum in eqn (2.2.25)actuallv contains the particular spia iwiee: once under its name Sç, andoncems oneof the possibilities irt mlncrn ing over S#. Equation (2.2.27) meansthat the total energy) S, is not equal to Ey Si, and a factor of Zz has tobe introduced in the mln!rn ation over the interactions, so that they are notcounted twice.

To the present approtmation, the exchange interaction between thespins hms thus tHrned out to be equiwlent to an interaction of each spinwith an efective âeld, Hç, which is non-zero even when the real appliedfeld H vauishes- lt gs essentially the assumption of Weiss, mentioned insection 1.2, whkh is why this e:ective field becxme known in the literatureas the Wr61.S molecular .#eu, or Just ihe Wreù.s .56ld. N' fore recently; thename of Wekss tends to be forgotten, and the nxrne mean $614 seems tobe taldng over. Under either name, the above treatment can be regardedat a certain level of analysis ms jut6cniion of the tmolecula.r fteld': whiclzWeiss Just postulated arbitrargly. To reach this level, we have postulatedarbitrarily the etstence of an exckange interacKon of the form describedby eqn (2.2.25). A deeper-level justiîcation calls for delvsng eqn (2.2.25)1om more bmsic pzinciples, which wm be done in the nead chapter.

The problem of ferromaaetism has thus been reduced to the problemof isolated spins iateracting with an applied feld, which is the problem ofparamagnetism treated in the previous section. It should onlr be noted thatthe energy here is .-.S .H, whezecs in the pre'dous section it wp.s -JpsS .H.Therefore, the argument of the Brillouin function in eqn (2.1.14) needs a,nappropriate normazization. When that is done, the a-component of Sz' isseen to become

sm(&z) = SBs g., , (2.2.2 8). where HL is defned in eqn (2.2.27).

Now the particular spin Sï is nntagged: on avemge there is no difere-acebetween that spin and any other which appears in the summation in eqn

18 MOLECULAR FVLD APPROXIMATION

(2.2.27). Therefore, b0th spins may the written without the index, ï or jtand'eqn (2.2.28) becomesj after substituting 1om eq.n (2.2.27),

(2.2-29)

which is a transcendental ecpation for det-lxnl'nl-n g (&). Actually, this rela-tion Ls not strictly defned when written in this way, because the summationover j depends on ï. It still calts for another aumption about this sum,the most usual of which is that JL:i is zero for spins which are not nearestneigizbours in the cz-ystal, and it has one universak non-zero mlue J forneatest neighbours.

lt Ls customary to use the notation

/ R ';N @-F J7y = M-=S )

9;iB$f9'J% = ' tQT

(2.2.30)

(2.2.31)and

(2-2.32)

where p Ls the number of nearest neighbours, and the squaze-bracketedexpression is for the above-mentioned assumption that spins which are notneazest neighbours have a zero excàange integral Other assumptions aboutsome fnite values for the next-nearest neighbours (of whic,h real cases ezdstA); or even further-away spinst are also possible with the snme algebra.An example wi.ll be #ven in section 2.3. For these notations, eqn (2.2.29)is

11. = BsLlö + aJz). (2-2.33)Older texts used to elaborate on graphical solutions of this ecpation, but

it is not necessary any more. With a modern computer a numerical solutionof ecm (2-2-33) is a tridal matter, and g e-an be plotted as a function of a forany mlue of h. For a rather small à,, thmis solution looks more or less like theplot in Fig. 1.% with slight variations depending on the mlue of S. Ia thelimit of vazdshing h, the solution looks approximately like the curve plottedîn Fig. 2+1. 80th these curves are plotted using aztother theory, which willbe discussed in section 4.6, and are not really plots of the molecular âeldappro-vimation. They ax'e shown at this stage only as a demonstration ofthe qnalitathe shape of the solution of eqn (2.2-33)- Howeler: for the caseh, = 0, which is the most interesting case for theorists, there Ls an analytic

2S2 7.

uz#j ,

FBRROMAGNBTLSM 19

O.8

I;,''> 0.6&Xhv

Q 0.4

r/meFIG. 2.1. An approkmate shape of the solution of eqn (2.2.33) for the case

lb = 0. The temperature: T, is normalized to $he Curie te'mperature, Tc,above whic,h the only solution of this equation is p = 0.

'

approfmation 25! to the shape of this solution, whic,h cxn be particularlyuseful when the molecular feld cuzve has to be computed many times, aza part of more comple,x computation, for exxmple in avera#ng over someparameter. Using the notadon

'm' = /,'(T)/?z(0), t = r/zc, (2.2.34)the xnxlytic approlmation Ls

(2.2.35)

Here a and b are adjustable paraceters, the best values for which aretabulated in Table 2.1, and

1c = 1 + . (2.2.36)4SLS + 1)

Table 2.1 lksts also the maMmum deviation of the analytic upression fromthe exact solution of eqn (2.2.33). This accuracy ks adequate for most prac-tical purposes, espedazly since the rnxxn'm nm of the deviation always occursfoz rather smallvazues of @: for which the accuracy Ls usually less important.

20 MOLECTJLAR FIELD APPROXLMATION

Table 2.1. The parameters c and b whic.b shouldbe used in eqn (2.2.35), and the matmum relavtive deviation, D, of tlds equation 'hom the solu-tion of the molecular feld theoretical relation, eqn(2.2.33), for diferent N?alues of the sph, S.

.6' a b D (%)1 -1 0182 x 10-2 O 26166 0.695; * *

1 7.7521 x 10-3 0.19270 0.708

(. 4.5249 x 1O-S 0.14825 0.5192 1.1241 x IO-S 0.11229 0.2771 -7.9838 x 10-4 0,080979 0.3562

3 -1.5270 x 10-3 0.052860 0.5351 -1.4780 x 10-3 0.027221 0.677

As has been mlptioned in section 1.2, the temperature-dependence oft:e magnethation in zero feld, as expressed by eqn (2.2.35) or as plotted inFig. 2.1, is expected to be valid i.a the infzrfor of tke rnngnetic domains. Inpractice, measurements are done in suKciently high felds as aze necessaryto remove those domains, and are then extrapolated to H = O in orde,r tocompare with the theoretical curves, such as the one plotted in Fig. 2.1.Details of thhs procv *1 be Tven in chapter 4.

Actually, if H = 0, there is no direction in space which cxn des.ne the z-axis which has been used for deriving ew (2.2.33) izï the 6mt place. In reMferromagnets there is no dilculty, because they are anisotropit, aad havea buîlt-in preferred spin direction. However, for readers who may wonderabout it azready at tlsis stage, it is suëcient to assume that the case H = Ois the end of a process in which a inite feld is applied, and then slowlyreduced to zero. Such a process is quite close to what is done experimentallyanyway.

2.3 AotiferromagnetismThe exchange integrals Jï.f were assumed in tke predous section to bepodtive, so that spins tend to align parallel to e>h other. TllLs positsvevalue Ls essentiaz for having a ferromagnetic order, but it is not necessadlyso in al1 materiazs, and exchxmge htegrals may also be negative. Actually,negative exchange coupling occurs in nature more ofien than a positiveone. When the exchange integral between nearest nehghbours is negative,it tends to align the neighbouring spins antiparallel to eath other, whichcan SSO give rise to a certain order at low temperatures. Such materials.do

ANTVEMOMAGNETISM 21

ezdstj and are cazed antfevromagnets.The thcory of this phenomenon was preseated by L. Néel even before

it was 6.rst observed e-xpezimemally; see the history as described in hksNobel lecture ($. He con<dered a coestalline material, made out of twosublattices, which are constructed in suc.h a way that the nearest neigh-bour of rw-vh spin belongs t,o the otber sublattice, with whic.h it interactsby a.n antifen-omagnetic exchangc coupling, -J, with J > 0. Interactionsqrith hrther-away spins have also been added in later studies, but here weconsider only a relatively simple case which is only a slight generalizationof the original Néel assumption of interaction betwee.n nearest neighboursonly. We assume that, besides the p nearest neighbours in the other sublat-tice, e.ac.h spin Jtlqcl interacts with p' neighbours within the same sublatticeby a ferromagnetic couplîng? -FJJ. Eowever, in order to maintaân a basi-cally antiferromagnetic case: we also assume that T << J. Hteractions withhrther-away spins are taken here as zero.

Other possibilities can azso be found in the literature, includng caseswhich carnot be described by such a simple subdivision into two sublattices.An outstanding example is Euse, which does have suc.h two sublattice,but with J = J$ so that it takc only a small perturbation (e.g. some

impursties) to change it fzom a ferromagnet to a,n autiferromagnet, or viceversa. We ignore these caes here, aDd do not try to study anything moregeneral than the specifed assllmptions.

Denodng the z-componeni of the spin in e.a,e,11 sublattice by Sz amdSa respectively, the Kective îelds on each of them are, according to eqn(2.2.27))

Ih = p/zuz'y'' + !?p'J' (-S ) - 2.gJ(x%1' (2-3-37)and

H2 = gyB.ïl + 2#.F(,%) - 2pJf%j- (2-.3-38)Substituting in eqn (2-2.28), one obtains

S r JJ + kp' J/ h'-S'; ) - apytxs'a ï, 1= SBs klps , tàs T (2.3.39)

and

(2.3-40)

For the case H = O everything is symmetzic, and ît Ls re-adily seenthat the two equatbns become the same by the substitution (&) = -(Sa)-Therefore, the solution is that the magnetization is the same for b0th sub-lattica, qnly in opposite directions, and each of them is a solution of

25 , /(S) = SBs -t (.p J -F pJt (S) , (2.3.41)

22

which Ls the ssme as wn (2.2.29), or (2.2.33), for H = 0-It is thus seen that in zero applied feld, the magnetization of each

sublàttice in. an antiferromagnetic material ls the snme as that of a fer-romagnetic material, with the temperature-dependence as in Fig. 2.1. J.aparticular, the order disappears above a certain transition temperature,whic,h is muivalent to the Curie point in ferromagnets, only in the caseof anxerromagnets tMs transition temperature is called the NQI imfnt!It must be emphaszzed, however, that a measureent of the total mag-netization, (Sï ) + (&), gives zero for H = 0- Thus these materiolq couldnot be discovered 1om the memsurement of the macroscopic magnetiza-tion, which is why their ecdstence was not suspected before Néel came upwith tids concept of two sublattice' and azz antiferromagnetic exGangecoupling. Nowadays) the antiferromagnetic order below the N;e,I temper-ature em.n be seen by neutron dxraction, because neutrons interact qeith

'

the loal magnetization when they pus tàrough the czy-stal. lt c,an alsobe see.n by nuclear magnetic resonance and by the Mössbauer efect, b0thof which measure the magnetic moment on the particulaœ atom in whicxthat nucleus is located and not the macroscopic magnedzation (see section4.1). The ex-iKtence of antiferromagnetism may also be inferred from mea-surement of the speclc heat, which will zmt be dl'skcuqe here, or from thesusceptibility above the Néel point, to be described in the next section-

MOLRGTLAR FVLD APPROXIMATION

2.4 The Curie-Weiss LawThe solution of eqn (2.2-33), or (2.3.41), is zero in zero applied âeld for anytemperature above a trnnvltion point, nxmely the Curie point in ferromag-nets, and the N#el poht in àntiferromagnets. J.n this region of T > To, itis Hown fl'om expeziment that all ferromagnets and antiferzomagnets be-come regular parnmagnets- Jt is p,lt4n quite clear from the foregoing, at leastqualitatively, that for a suKciently high temperature the thermat Quctua-tions overcome the exeangeinteraction between the spins, thus eliminatingthe ferromagnetic or antferromagnetic order and maldng the mate-rial asdisordered as a paramagnet.

For the quantitative study of the high-temperatme region, we will con-sider together the case of ferromagnets and of aatiferromagnets, becausethe algebra is esseatially the same for b0th, and there is no point ia 1m-necœsary repetitions- Actually, we take the ferroma&ets to be a particularcase of the two-sublattice anliferromagnets, as deîned in the previous sec-tion. For a true antferromagnet we assumed there that J and JJ are b0thpositive and that J? m J. But we can also include the case of a simpleferromagnet as the particular exq- J = 0, (i6',) = (5'a), and Hï = H2. Itmay /3.rstn be obtnlned as the particular case J < 0, J' = 0.

In the high-temperature region it is suëcient to approvirnate the Bril-louin function by the frst term of a power serie expansion, as in eqn

TFR CUOWEISS LAW 23

(2.1.20), because the argument of lhis function is always smatl. Therefore)eqns (2-3-39) and (2-3240) may be replaced by

.b'(S + 1) , , a yfsgj) (z<.<a)(&) = (gIJnH + 2, J (Sz) - 1)3ksT '

and

S ( S + ). ) , ,(&) = k y. Lgp.v.H + 2, J (,%) - 2pJ(&)) ,3 s (?-4-43)

wkich are two linear equations ia the magnetizatons of the two sublattices-It is not diEcult to solve such a set of two equations, but it is not even

necessary to do so, because it is suEcient to nHA topct/scr the two equationsand solve for

C ff 0-'tslottch = (St) + (Sz) = - - (.%oq;x:)a (2-4-44)gISBT T'

where

zsLs + 1) zc = .(Jp,s) ,3ks

Therdore, the total magnetic moment is

2S(S + 1) ? t a 4 45)O = (.pJ - .p J,) . ( . .31s

(2.4.46)

This tnzn perature-dependence is known ms the C'ttrie-Weiss Ic'tp. lt shouldapply b0th for fGromagnets and for antiferromagnetsj and indeed it îtsexpezimezns on b0th types, with some exceptions (7) which are igaoredhere- lt is more usually expressed in terms of the initial susceptibility,

CH(Ma) = #;zB(&oza:) = z +. a.

(Mz) CA .t! --Xinitlal = 'XVR U 'x-.z H T + O (2.4.47)

which is more general than the relation i.tt eqn (2.4.46). In many cmses theseexpressions are equivalent. But sometimes the frst-order approximation toBs used in eqns (2-4-42) and (2.4.43) is inodequate for large felds, .ï1, andit is necessary to apply the Curie-Weiss law only for small H-

In bolh ferromagnets and antiferromaaets the use of this linear ap-proimation is certaimly justifed (at least for small Hq above the Cuzie orthe NV temperature, becamse sn both cases, (Sz/ = O for H = 0. Sincewe have seen that the ferromagnets are the cmse J = 0, it is clear 1omeqn (2.4.45) that O < 0 in thœe materials. For antiferromagnets we havemssumed that JJ << J, and eqn (2.4.45) yields a posité'tm 0- .

MOLECULAR FXLD APPROXIMATION

lt ks thus possible to tell the dlFerence between ferromagnets and antî-ferrcmagnets from the measurement of the initiltl susceptibility a'bove thetransition temperature, when they areboth pvxmagnets-rrhe Cuzie-WeeLaw in these two types of materials is shown soematically in Fig. 2.2 andthe deerenœ is obdous. Wiq property is pmicularly useful for materi-azs in which the Curie or NV poini is at a very 1ow temperature, whichis not easy to acces directly. In such a case it is prva-ble to determine9om the high-temperature data whether the matprixl ks going to becomeferromaretic or antiferromMnetic at low temperature. If there is no tran-sition at all, and the material remnlns parnmxgnetic down to absoluta zero)J = J' = 0- = 0 and the reciproe of the high-temperature susceptibilityshould exdrapolate to 0 at T = 0.

At lower temperatures; the linear approvlmation to Bs is not adGmate.However, it vnn.y still be ltlr.pz' at the transition temperature itself, wherethe disorder just begins, because the functions are contixuous and a f nite(Sa) for S = 0 must start Trom small mlues. Therefore, ct the transitiontemperature, the set of linear equations (2-4-42) and (2.4.43) should stillhave a non-zero solutîon for (,5'z) and (S2) when S = 0. 'Phe coadidonfor such a solution is that the deteminant of their coedents vznishes,nxmely

1 .- C*p'Jl

6'*yJ

C*pl 2S(S + 1)C* = .3t-sTw (2-4-48)

HenceTx = (2.4.49)ZSLS + 1) , ,@J+ p J ) .3:8

Comparing wiih eqn (2.4.45), ilds resuli means that Tc = IOI for asimple ferromagnet with in'tarArrkion between nearest ndghbours only. Sim-ilxrly, Tx = 0- for a Kmple autiferromagnet with interaction between neanest neighbours only, nmely for the cae J? = 0. If TN # (% the diferenœbetween these experimental qmlues may be taken as a meuure of J', pro-vided of course that the dl'Ference ks rather small, so that it fts the baskcassnmption of this calculation, that T <<t J. The value of 0- is always usedwith eqn (2.4.45) as an experimental eeuation of the Gchange integra),J. At 5.rst sight is seems as if the Curie-Wee 1aw also contes aaother pa-rameter, C, whkh exn be compred .1t,h oemOt. Howcvec in practiœthe number of spins in the crystzl ites of ferromagnets is not = integer,as will be dismzssed in ihe next cxapter, and the susceptibility data mustbe nozmalized. Therefore, the parameter C does not contasn any uxfulHormation, at least for metals.

It shoid be particularly emphasizod that thœe reult.s apply strictlyto cases wltich ft the assumpuon of two sublattices, witk predominantly

TEE CUOWEISS LAW

nearct-neighbour interaction. Other cases may be studied in asimiqar way,but do not necessadly lead to similar xesults. J.u particmlar, it is very easy tcsubdivide a simple cubic, or a bcc? into two such sublatdces, azd have thenearest neighbour of each lattice point iu the otltcr sublattgcel bui it justcannot be done for a.s fcc lattke. An fcc Iattice is more readily subdividedinto jonr sublattices, in which the nearest xeighbours oî eacll spia are ictall other three sublattica. Speclcally, one sablattice contals tke points(k, 1, ,rzl), one contains (k + la :2 + 1-2 , m), one contains (k + l.a , Jj m + j ), and

i for integral k, lj and m. Numbermg theone contains (k) J + lz t m + y))sublattices in this order, counthg the nearest neighbours, azld using thehigh-temperature approx-imation for the Brillouh function, the equationsto be solved asre

s(s + 1)(,6'z) = LgP'BH + 8J(t,5'a) + (,$'a) + (,$.4))) ,zksr (2-4-50$

s(s + 1)(,Sa) = g. LgpeH + 8-7 ((&) + (,6's) + (,S4.))2 ,3,+ (2-4.51)

s(s + 1)Lszé = Lg>BH + 8-7 ((5'z) + (&) + (&))q ,3ksT

s(s + 7.)(&) = k z LgP'BH + 8-/ ( (5'z) + (&) + (&))) . (2.4.53)3 B

By adding together the frst two and the last two equations, it is seen thatthere are actually two sublattices,

(Sï) = (,5:.) + t.Sa), t,SIr) .'.= (u$'z) + (5--k) à (2.4.54)

(2.4.52)

with t'qro equationsr

2SLs+ 1)(q'h) = aur L9IJ'BH + 4.7 (f&) + 2(SrI))q

=dzsls + 1)(,S11) =

asz LgpeH + 4./ (2(,6'z) + (,S11/)) -

The fcc lattice has thus been subdivided into tèo sublattices) but witha number of neighbours in the same sublattice being dxerent f-rom tltelrnumber in the other subhttice. This slight (liserence does change the fore-going algebra Repeating the same calculations for the initial susceptibilityleads to the same Curie-Weiss law as in eqn (2.4.47), with

8J5' /u5' + 1)o- = - . (2.4.57)ks

(2-4-55)

(2.4.5é1k l

26 MOLECULAR PIELD APPROXWAXON

I -,QN

Tempemature, T

FJG. 2.2. Schematic representation of the initial susceptibiiity of an aati-ferromagnet: above and below the Nle) temperature, TN, and of a ferro-magnet abo've the Curie temperature, To. For T < TN, a singl+crystalaqtiferromagnet may hap-e diArtmt parallel, xjj , and perpendicala'r, x-u,susceptibilities- 5br a ferromagnet, x has no meaning below Tc.

Repeathg the same calculation for the transition temperature y'ieldsTc = - 0- , for a ferromagnet, fz. when J > % but T.v = O/3 for anantiferromagnet, ï.6. wllen J < 0.

lt Ls also possible to use a non-zero Eeld H in eqns (2.3.39) and (2-3-40)1solve them numericalls and fnd the initial susceptibility of an antifenomagnet èe.kzlr the N&I temperature- For these two muations ms writtenhere, the result is

(f/tB )2xkaqtzal = j g ) (2.4.58)J'

which does not depend on the tempervture- However, cry '

e materialshave an anisotropy, which r-ein be expressed ms an internal Geld that prefersthe magnetization to be oriented along certaân czystallorraphic axes. De-tails will be Sven in section 5-1, but it may already be realized that thez-direction which %'e have mo,ed for the quaatizatfon direction may be con-nected with the crystallographic axes, aud is not necessarily identical withthe direction of the applied feld) H. In such cmsesj the susceptibilîty in eqn(2.4.58) is the JCZUJJeJ susceptibiiits xjj, in which the small applied H isin the saae direction as that of the internal îeld. One may also appiy a

FIORTMAGNETISM 27

small feld, H, perpendicular to the direction of that internal f eldj and tom-pute the perpendicmlar initial susceptibility, x.t., obtaining the temperature-dependence shown seematically in Fig. 2.2. H prindple, b0th x(l and x.s

. may be meazured in a single-crystal sample.H practice, this measurement îs not that simple, because even single-

crystal samples are often subdivided into domains magnetized along diser-ent crystallovaphic axes, and one cn.n only measure some average betweenthe curve for x(( and x.z in Fig- 2-2. For exltrnple, in a cubic material withan equal probabzlity of domains h each of the three axes, the measured xwill be lxj! + 2aX.s. In a polycrystmlllne sample one almays meazures justaan average of these two curves. Thereforel the details of these curves donot have much use in comparing qrith experiment- However) because b0thcurvK have a discontinuity in the derivative of x ns. T, aay average orcombynation of them will have such a discontinuity. Therefore, the positionof this discontynuits or cusp, in the experimental data is an accurate mea-surement of the Néel temperature, Tx. A more accurate meazurement of

euTx is obtained from the anomaly in the specifc heat, which is beyond thescope of this book.

J.r. a ferromagnet 0- < 0 and) according t,o eqn (2.4.46), the susceptjbil-ity diverges when the temperature approaees Tc 9om above. 'IWu-q :'nllnA-tyis just a manifestation of the possibility of having -%% # 'B for H = 0.It is tempting to desne a ferromaret as the 15n11 t of a paramâgnet whenx -+ co, but such a dvnition does not have an;r moajag. As has alreadybeen mentioned in section 1.1, the relation (1.1.1) is not G4lflled in ferro-magnetsl aad Mz is not even a unique fundion of H below Tc. For the samereason, there is no point in using a small H in eqn (2.2.29) to calculate asusceptibiljty below Tc, in the same way as in an antiferromaaet, yt doesnot prevent theorists from calculating it anzway, but such a suscepdbilitycannot be me-asured because a large feld is needed to remove the magneticdomains (see sectioa 4.1). In principle) the Mössbauer efect (which mea-sures the magnetization inside the domains, as calculated in thl'q chapter)can see the Xect of a small applied feld. In practice the accmracy of theMössbauez esect is not sulcient to see even the Gect of quite large felds.Therefore, the susceptibility below the transition temperature has a mean-ing only in antiferromagnets aad not in ferromagnets, aad this is the we.yit ks shown in Fig. 2.2.

2.3 FeuimagnetismThe emxac't cancellation of the opposite maaetization in the two sublatticesis possible only for identical magnetic moments in b0th lattice points- NéelA considered also the case of two sublattices in which the magnetic mo-ments are not the same. It happens either because they are made of atomsof diserent materialsj or because the f.ems are not the same, for exp.mplewhen there is Fe2-F in one sublattice aad Fe3+ i.n the other. In such case.s an

28

antiferzomagnetic coupling between the two sublattkes leads to a pactiako'mcellaéfm of the magnetic momelt. Tbe resulting total magteti'zationat 1ow temperatuzes is the diference between that of the two sublattices,which is not zero.

Some of the mateziazs to which such a theory applies were actuallyknown before Néel came up with the theory, but they were thought tobe of the same clas as fezromagnets, whic' h confttsed the issue. lt maybe interesting to note that the oldest permanent magnet, knomra alreadyin ancient times, is magnetite, FeaO4, which is a ferrite. As such, it is afezrimagnet, and not a ferzomagnet, according t,o the classifcation of Née,l.It Ls also interesting to note that the molecular feld theory of ferrimagnets,as presented ilz this sedion, fts very well (6q the temperatnre-dependenceof Ms in magnetite.

The nzkrne Jcrrfte,s was irst given io certain matetials madc of ironofdes together with some other ofdes. Néel used this aame as the baeisfor the class of matezials whlch he called ierrimagnets. This name f.ts onlythe Frenc.h pronundation, but it was also adopted in Engltsh, even thoughit does not Gt this language. ln spoken English the diference betweenfenomagnetism and ferrimagnetism e-q:n be heard only when the empha-sisis put on the wrong syllable.

We start by generallca-ng the theory of antiferromagnetism in ihe high-temperature region, for which the Brillouin ftmction can be approfmatedby the linear term. We considcr the snmne two sublattices, with mrcAangcinteraction -, J between nearest neighbours only, and assume that thereare p of them, all tn the other sublattice. Howeverj here we do not takeall atoms to be the same, but Mqlgn diFerent quantum numbers, S1 and% respectiveelyj to the ions on each sublattice. We also use diferent g-factors, gk and g1t for the two sublatticcw The genernlîezation of the hig, h-temperatare equations (2.4.42) and (2.4.43) is straiglttfozward, leading io

MOLBCULAR :&LID APPROKIMJATTON

sz r. sï + 1 ,h(S'y zb, = - ' lgl>ps.à.f - 2.p.7t..%z))31-,a:2-.-

S2(Sa + 1)(Sca) = aksz Lg1#BH - 2pJ(Szz)) .

(2.5.59)

and

(2-5-60)Using the notations

:2/4 SjLsj + 1)c = -f- . --,i 3:s2pJe; = j sqls. + 1), (2.5.61)

for ï = 1j 2, these equations become

CïHT J S . o ) + 0- 1 L S a z ), = ,$ ...#-x liB

L'-CZ':J0- c ( S'j z%) + T (Sz z h,. = . (2-5.62)9211,1.4

F'ENNTMAGNETJSM

The solution of this pair of equations is

(2.5.63)

for i = 1, 2, where the index % has been omitted for simplidty: it beingtmdezstood that the averages are those of the z-components. Therdore,the (initial) susceptibility is

29

S.pe (&) =

z o y ag QT - Cz C;O- 1O- zh ,% T- y

2 ,2. ! (C$ + C1 ) T - 2 C : C, 0- 1 0- 2x = - gLp.sj%.? = .

a o. o- a ' (2.5.64)H . T - zz = 1

This susceptibility diverges at T = Tc: where

T = v' ' -----0 O ( 2 . 5 . 6 5 )c l 2!

xs is the cxase in ferromagnets- Here, as in ferromagnets, Tc is the Cnrietemperature) above whic,h the magnetimtion is zero i!l z,e1-0 applied f eld.The latter can be seen, as in the predous sedion, by look-ing for a non-zerosolution of eqns (2.5.62) with H = 0, the condition for whic,h is

T c 0- l = O j0- a T c (2-5-66)

whose solution is eqn (2.5.65). It is =ot a coincidence. The divergence ofthe suscepfibility, in ferromagnets or in fezrimagnetsj oziginates fzom thevanishing of the same determinant in the denominator which appears inthe solution of the simultaneous linear equations.

H the case of fezzimagnets, the temperature Tc is caxed thlf Azvi7zzc<-netic Ctlde noini. Above this temperature, 1/x is the curve given by eqn(2.5.64), whicN is obviously not the straight line of ferromagnets. However,at much higher 'temperaturK, T D> Tc, eqn (2.5.64) becomes

1 > T+e-0 R tx (Ct + CcIT (1 - e/T) C3 + Ca (2.5.67).

where2 z Cc To

0- = . (2.5.68)Cï + G%The asymptote is thus a straight line which cuts the temperature n='q at anerative xralue, T = -e, as is the behadour of a,n antiferromagnet (see Fig.2J3). Tlds 0- is called the lmvamqgheiéc Ctlrl: yoént. lt is usually diferent9om Te) as can be seen from eqn (2.5.68). This equation actually meansthat 0- = Tc only when Cï = 6%.

30 MOLECWAR FVLD APPROXIMATJON

! W

. . z2A

#'z'

#Z'

.# I.

-'

j.'e/' 1

.' J.'

Aé'

A'.W

-6 TQ

l'emperattlre. f';Fï(7. 2-3. Typical behavîour of the imtial susceptibility of a ferrimagnet

above the ferrlmagnetic Curie temperature, Q, as computed 9om eqn(2.5.64*2.Equation (2.5-67) is the Curie-Weiss law. We have thus seen that this

law is obeyed asymptotically in the case of ferrimagaets. In these materi-als, the high-temperattre data for the susceptibiity look like those of anaatiferromagnetj when titey are utrapolated to lower temperatures. How-ever, when the temperature approaches the flwsmagnetic Curie point, Tc,the susceptibility of a ferrimagnet looks like thaà of a ferromagnet. BelowTe there is a spontaaeous, non-zero magnetization ilz a zero applied feld,which is.also similar to a ferromagnet-H that temperature re#on, a suscep-tihity exn also be calculatedl but it does not have any physical men.ningbecause it cannot be measured, as ks the case in ferromagnets.

For T < Tc this liztear treatment breaks dqwn, and non-ljnear equationssuch as those of eqns (2.3.39) and (2.3.40) have to be solved numerically.Sivce the temperature-dependence of the magnetization Ls not the same fotthe two sublattices, the results are usually more comple.x than in the Zmplecase of antifcromagnets. J.u particular, it mayhappen that sublattice .?1 hasa much larger spontaneous magneeation, but a smazer Curie temperaturethan sublattice B. At 1ow temperatures the total magnetization Ls parallelto that of a'1, but at higher tempezatures the magnetization of sublattice..4 vaYshes; and the total magnetization is that of sublattice B, whichmeans that it is oriented in the opposite direction. Therelore, there is atemperature in between, known a.s a tompenaztion pofnt, at which the

E'EPAIMAGNETISM 31

total spontaneous magnetization passes f/lrptœ?i zem before reaching initev'alues again. r.:'.L.?.S phenomenon has indeed been observed in some ferritesA of the general formula FesMaola, where M is a trivalent rare-earth ion.

' Many experimental data on all soris of fezzites can be found ix books onthis subject g8j.

The Jfne.r study for the temperatures above the Cmie point can beeazily œxtended to more than t'wo sublattices. It is readily seen fzom theforegoing that the more rneral form for eqns (2.5.59) and (2.5.60) is

(2.6.6 9)

for any number of sublatuces. Using the notatiox of Ci as in eqn (2.5.61)Md

?JLL .0- ç.f = y/ S4(& + 1)S#(z% + 1) , (2-5-70)3/v1and replacing the average spin component by the variable

.t.?'.i/.$: h.1:2 = - (siv ) , (2-5.71/i

the set of equations (2.5.69) becomes

Tz: - 0- çj'zju = AH. (2.5.72)J

For the solution of this set of ïmulianeous line.ar equations R is conve-nient to use matrix notation. Let 9 be the matrix whose components aze0- ç.o and 1et I denote the unit matrix. Let x be tlle vector whose compo-

1/2 be the vecior whose components are ç. Thenents are zi, and let Cset of equations (2.5.72) îs then

7.,/2.s.(Tf - 0)x = C , (2.5.73)whose obvious solution is

x w trz - 4)-1c'/2s (2.5.74)The susceptibility is given by

1 1x -- - k'K z = - x ï A ,Jf Hi. L

(2.5.7 5)

and this sum is the scalar product of the two vectors de:ned above. Sub-stituting for x *om eqn (2.5.74),

32

(2-5-76)

This susceptibility diverges wben T equals any one of the eigenvaluesof the matrix 0. There are materials, such ms MnO, which have severaldiferent antiferromagnetic patterns, exprcsed by dferent eigenvectors x.Trandtions from one structure to another occur at the temperaturc forwhich the eigenvalues of 0 cross each other. From the measurements ofthese transitions it is possible to extract the wlues of the Jfj which appeari'n the deflnx-tion (2.5.70) of 0- ii. R is also clearthat the Jcrlasà eigen=lue of6 is the ferrimagnetic Curie temperature, if the material is ferrimagnetic, orthe Néel temperature, if the matua,l is =tsferromagnetie just below thattransition.

Expanding x of eqn (2.5.76) in powers of 1/T, it is seen that at hightemperatures

MOLEGJLAR FXLD APPROXIMATION

1 1..a 1 ,.a -1 .j,Fax = C) '' . x = C .' . IFJ - é) C .#

1J2 J c1/2 . c1/2C . g - -'j -.ac1./a r:a ,X= T F T-O '

/(2-5-77)

wherec l/2 . yc 1/2 E 0- çj CjGji.i z ,5 ,s)()- = . w = m ( . .c1/2 . cl '/ L5 Eç tpy

is the plamagnetic Curie point, as defned in the foregoing. Wdttea ex-plicitls

2 Egj XjSILSi + 1)&(SJ + 1)0- = . (2.5.79)31 Sy S2(& + 1)The eigenxalues of the matrix 0 are often (but not always) related in

some way to those of a similar matrix whose eigenvectors represeat theordered confguration at zero temperature. Details are beyond the scope ofthis book and eltn 1)e found on p- 123 of the treatise of Anderson (9j.2.6 Other CasesThe main advantage of the molecular feld approfmation is that it is muchssmpler than any other approfmation to the quantum-meehanieal Hamil-toaiau izï eqn (2.2.25). This advanvge is a vez'y important one, becausea simple theory can be extended to indude more complicated additions,which a complex theory cannot.

For those who are only interested izt a better approfmation to certna''nparts of Rg- 2.1, obtaled from eqn (2.2.25)) thère are inded methods toachieve it, as v'ill be seen in chapter 3. Eowever, these methods cannot gobeyond eqn (2-2-25), wkich may sometimes be an insuEcient approfmationto the physical reality. Tlze molecular feld approfmation is sufdentlysimple to allow for other energy terms to be added to that equation, whîch

33

is not usually possible in the more sophisticated methods. H these, eventhe feld 1:1 is often dropped from eq.n (2.2.25) before anything of interesthappens. ln many cxq:x-q the additional accuracy of the other methods ismore than ofsetbr thereduced acmzracycausedby neglecting enerr termswhich are not really negligible.

One example is the magnetocustalline anisotropy enero', which willbe defned in section 5.1. 'nis termj and the biqnadratic ezchange whichis (101 an enerr term of the form (S1 . Sa)2, are very easy to add to eqn(2.2.25) under the molecular feld approfmatiop but not so exsy with'anyother method. The latter energy term has become rather popular recenûy;because it is quite strong 5:11 multilayers. lt has been sho'wn (11J that itcan arix 9om thickness êuctuations in such flms, or 9om magnetostaticinteractions due to (12) stzrface roughness, and indeed it has been obseawed(13) in many estems. A more outstanding example is t:e anksûtropic eg-change which is an energy term (14) of the form D . (Sl x Sz). It tries toarrange spins pevendicnlar to each other. Obviously, such a term cannotefst in high-symmetry crystals, because if S1 aad Sz cmm be interchangedby some smmetry operation, the vector product changes sign, and D mustvanish. However, this term does nviqt in some low-symmetry antiferromag-netic crystals, and 'by competing with the exchange it causes the directionof neighbouring spins to be slightly oq the exact antiferromagnetic direc-tion. Therefore, there ks then a net spontaneous magnetizatiow called meakJerrozntzgnetïdrp,, in a direction perpendicular to the antiferromagaetic nxis.This case is given here just as an e'rnm p1e of those which can easentiallybe treated theoretically by the molecular feld approfmation only, unlesssome trivial form (15) is involved.' The f4111 details will not be #ven here,but they can be found in the review of Moriya (16) oz in the later literature:such as the study of orthoferrites (17) by Trevu.

As a lgst evxm ple of the usefulness of the molecular feld, we considerthe case of an impurity, non-magnetic atom in a ferromagnetic lattice. Forsimplidty of drawing we take a tmo-dimensioncl muare lattice, as shownin Fig. 2.4, where the ceatral atom in the Egure does not have a magneticmoment. The problem is essentially described by eqns (2.2.27) and (2-2.28),which we use here with the notations of eqns (2.2.30) and (2.2.32). For thecase of interaction between nearest neighbours only, and H = 0) theseequations are

F,i '= Bs 1(T) /&j , (2-6.80)where the summation is over the nearest neighbours. Specifcally, dening(shells' of distaace from the impurity atoml az in Fig. 2.4j and considedngthe shells next to each atom ms in that Egure, we can write

p'z = Bs ((2#a + pslûq n (2.6.81)

OTUR CASES

34 MOLECULAR EVLD APPROXTMATION

(lls (()* (1)3 (1)* (lls(è)* (è)a (è)1 (è)c (è)*(l)s (()1 ip (()1 (l)s(è)* (è)a (è)1 (è)c (è)4(y), (y). (y)a (y). (y),

F1G. 2-4. Shell numbering around an impurity 'atom în a twœemensionalsquare lattke.

p,g = B s ((25: + 2/.z4)a) , (2.6.82)etc. These equations rzul be cnmn-ed up to a shell which is considered farenough 9om the impuzity for its magnetization to be that of a pure feaa'ch-magnet. Numezical computations em'n 'then t'y to include an extra shell andsee if it makes an appreciable diference, and add another one if it does,until convergence is achieved. This problem was solved (18) as outlinedhere for a three-dimensional cubic cystal. It is m'ach more complicated toapproach it by aay other method.

The molecular feld approzmation also has the big disadvanvge of be-ing too simple. It ignores the actual thermal fuctuations and the possiblecorrelations between fuctuations of neighbouring spins. Tt can thus be e.wpected to be a good approfmation only at rather high temperatures, wherethe disorder is high. At low temperatuzes, one rxn do much better by con-sidering small excitations of the state t'a whic,h all spins are aligned. Thisappremation, kaown a.s spin ocre's, xzill be described in the next chap-ter. Then, when the temperature is nt511 higher azzd approaches the Curietemperature, thermal îuctuations become correlated over the whole lat-tice. In this critïcftl region the best accuracy is achieved by the so-calledrenovmalinabion. prœzp, whicil will be discussed in chapter 4.

THE HEISENBERG HAYLTONIAN

3.1 Spln aud Orbitln the previous chapter we have azways refez'red to the maaethation of thespin at the atomic lattice sites, which is just a marmer of spealdng. It isnot vez'y accurate for some ferromagnetîc materials tu whic.k there is also aconsiderable contribution âom the electron orbits. However, for the level ofthis book the dxerence is mostly semanticp because we consider the totalmaaetic moment of eac,h atom, for wkich we have ttsed the spectroscopicsplitting hdor (also known as the Landé g-factor),

(3-1.1)

which rm.n be used in all cases. For a pure spin, L = 0, J =' S$ one obtainsg = 2, which is twice as much as for tke case of a pure orbit: S'= 0, J = L.

The most direc't meamzrement of this factor g for eac,k material is bya method Hown as the Einstein-de Nbxq eoeriment. Iu l13iK experiment,the sample is suspended ia suc,h a way that its angular displacement eltn

be measured 119) to a bigh accuracy. A magnetic âeld is applied to changethe sample magnetizatâop thereby cllanging the a.:1511% momentum of theatoms, thus causing a certain rotation of t:e whole sxmple. The value ofg is then obtained &om the ratio of the change in marethaton to thechange ill the mechanical angular momentum, and rm.n show which part ofthe magnetic moment is due to the orbitaz contributiom It turns out thati.n most of the common ferromagneà the orbital contribuuon is negligiblysmall. The reason is that the electric Eelds in the lattice tau'a the plane ofthe orbits into czystallographic directions, thus mab-ng Lz average to zero,or at leazt to a small number. In some rare earths there is an appredableorbital contribution) and to include them properly we should have aciuallyused a magnetic moment of glnns at the lattice site. Howeverj these casesq'ill be ignored here, aad we shall just take the magnetic moment to be;& = ggBst where Jzs is the Bohr magnetonr as in chapter 2.

It is interesting to note that the orisnal experiment of Einstein and deHaa.s was done with a poor accuracy and 1ed to the wrong condusions. Itwaa repeated by others who dared not publish their (better) resull (20)because of the high prestige of Einstein's npvne: aud it took years 'before itwas established thai the correct g ia Fe is nearly 2, and not 1.

J(J + 1) + S(S + 1) - LLL + 1).ç = 1 + ,2J(J+ T)

36

lt is worth noting that the spin responsible for ferromagnetism is thatof the d shell i.n the transition metals, and the f shez in thc rare barths. Fora flled shell, the total spîtt is known to be zero, and these shells are uzkfzlled' in the above-mentioned materials. The conduction electrons of the outershell in 170th casu are not bound to their atoms in the solid state, but are1ee to move and are actually shared by the whole czystalj as is the case inc!J metals. These itineraat electrons do carry with them some interactionbetween thelocalized spins at the lattice sites, as will be discussed in section3.4.

3.2

THE HEISENBERG EAMILTONIAN

Exclmnge InteeractionBesidu the indirect interaction cvried by the conduction dectrons in met-als, theze is a direct e'xchange interaction between spins of the ions at thelattice sîtcs: b0th in metals and in insnlltors. It has no classical analogue,and is caused by overlap of the electronâc wave functions in quantum me-ch=ics. It is this part which is discussed i!l this section.

Consider a system of N electrons which are bound to 25f atoms. Letthe eigenfunctions of a.n electron bound to atom No. 1, when the latter isisolated fzom the rest of the system, be dcnoted by bhLpïj, where pï areall the coordinates of that electron, includiag the spin. Since all the atomsare identical, if atom No. 2 Ls separated from the others, an electron boundto it wi)l have the same set of eigenfunctions: on'ly at diferent coordinates,namely mLpgj, and the same applies to all the other atoms.

Supp:se that the M atoms start from a position where they are wellseparated 9om each other, and then they are pushed towards each other.When these atoms get closer together, the single-atom levels start to be-come mhed. However: when this mifng just starts, there must st521 besome relation between the energy levels of the whole system alzd those ofthe separate atoms. ln particttlar, for atoms at a very large distance 1omeack other, the energy levels are

Mu = 1J(2& + 1)

ï=1

times degenerate, if Sg is the spin of the f-th atom. This degeneracy isremoved when the atoms are nearer together, eac,h level being spDt into uones. We assume, however, that the atoms are not yet very close together,an.d the hteraction between them is sneciently small, so that this splittingis still small compared with the dist=ce beiween the Hlperent originallevels. In such a case the origoal, atomic levels are still distinguishable inthe whole spectzum.

It is felt htuitively that in such a case there shottld be a way to buildthe eigenfnmctions (or at least a reasonable approrimation for them) out

EXCHANGE WTERACTION 37

of the functions vilspg), even though it Ls not so eny to justify such afeeling mathematically) cr even to state tke conditions for it by a rigorous,mathematical defniton. The simplest combinatbn whic.h can be built upfrom t/?i(py) is the product

'/ = m (#z) W2(#2) - - - WATIA-v) , (3.2.2)aad its permutations. However, this funcdon is not allowed, because itdoes not obey the Pauli exclusion principle, not being antisymmdric tointerchanging two of the electrons. We must take a linev. combination off anctions of the form of eqn (3-2.2) t,o achielre the necessary antisymmetryjfor which it is possible to use the determinant

detgwk)# = . . , (3.2.3)

where detlwkla Ls a notation for

pïLptj t#l(pz) wllps-l '

wat#tl eacpa) tntpx) ;detlywl = E -

!E )wxt/hl w-vtpal - - - vxlpxlt .

(3.2-4)

interchanging any two electrons is equsvalent to izltezchangiug the positionof two columns in the determinant, which is knowa to Teverse the sigmTherefore, the form of eqns (3.2-3) and (3.2.4) is iu accordance with thePauli exclusion prindple. '

E( wi'll not try to proent the most general cmse, and will just assumehere that the set of fnnctions w: is an orthonormal set, even though theconclusions whick we are going to draw can also be proved under lessrestzictive conditions. An orthonormal set is one for which

wzlpzlybtpzld'rz = 81.13 (3.2.5)

where the asterisk designates the complex conjugate, and 8 Ls the Krœnecker symbol, which is equal to 1 when ï = j and to O otherwise. '-theintegration in eqn (3.2-5) is over ûll the coordinates in pz, namely an inte-gration over the real space and a snmrn ation over the 1wo spin doordinatesof the electron. in practice, the assumption is that wf (p:) = /:(rz)qç((z),where rz are the space coordinato, and 'ryç are normalized functions of thez-component of the spin, which may be named Cspin up' and tspin down'.The latter are always orthogonal, so that our mssumption means that (%clso make an orthonormal set of functîons.

We also assume that the elcctrons of the inner shells are tightly boundto theiz nucleij and only the wave functions of the electrons in the outer

38

shells are Wected by the iateraction with electrons of the other atoms.Obviously, the more shells that are taken a,s the ùouter' ones, the moreaccurate the calculation ié, but usually it is not praetical to extend thecomputations to more than one or two shells. Wken this technique is usedas the so-called Hartree-Foc.k method for computing wave functions, onerarely extends the second g'roup beyond the valence electrons-H any case,the inner electrons together with the nucleus are taken p.s aa ion, whichcreato a certain potential at the position of the i-th electron. The potentialsdue to a11 tke M ions add up at the position of the Fth dectron to a cluewhic,h we denote by Tl. The Hamiltonian of tke system of N electrons isthen

N N z1 ? eX = lh + --' Y'Y + 'l'o , (3.2.6)2 '=' riif= T fy#=l

THE EEISECERG HAXTOMAN

where '/fc ks the Hamiltonian operating on the ion cores, rj# is the disfaacebetween electrons i and #, the prime over the second sum eliminates thecaeoe ï = j from the summation, and

25, .a'/'ff = - Vç + Tl .2zn,c tk3-2-7)

Here Vï operates on the coordinates of the Fth electron.Ushg this Hamiltonian and the eigenAlmctions defqed in eqns (3.2.3)

and (3.2.4) the energy of this system Ls

1 +*1 = /*S/ dn da - . . drN = det-kslsdetkk) dTz dm ' ' . drs.Nj .

(3.2.8)Since the operator is linea'r, the integral rxn be written as the sum ofintegrals over 'the various terms, namely

N NR zzr = Si + - th.i + Ec,2f=l f,J=l(3.2.9)

wizere we have def ned

(3.2.10)

(3.2.11)

(3.2.12)

EXCHANGE DTEMCTION 39

The last term in eqn(3.2.9) involves only the ion cores: and does not interestus here for the study of the electrons. Therefore, we want to evaluate onlythe frst two enera terms.. Let us consider frst the Ei term: and recall some of the properties of adetermhaat. It has X! termsl each of which is a product of N wkls. ln thelatter product there arenevertwo fanctions which are taken 1om eitherthesame row or the same column. Therefore, '/tç, which contains derivativesonly with respect to the particular coordinates p6t operates only on oneparticiar wk Lpjj in the above-mentioned product. The other terms m>y bemoved to the left of 7Q'. Integrating over the coordinates in those terms, theintegral vanishes according to the orthogonality assumption in eqn (3.2.5),unless the determinaat of w:, contains the comple.x conjugates of all theterms of pk which we moved to the left of '>ff. There is one and only onesuch term in detlrglzj for evezy term of detlwk) whidt fuKls this conditionand all the other terms integzate to zero. Usually, determinant terms maybe positîve or negative, but here each such term in one determinaut ismultiplied by the same term in the other determinant, so that the productis always poàitive. The integration over the terms moved to the left is then1, according to eqn (3.2.5), and we are thus lefL wîth the intevation overpo which is the only term which remains to the right of ?Q- In other words,Si is made out of a sum of X! terms, each of wikic.k has the Jérr?z

. e h r.t,.j ( j tjy.W k t # i z ï V k s V ï , : '

For given k and ï, the function vkLpj) appears in LN - 1)! terms of thedeterminant of eqn (3.2.4)- Htegration of each of them leads to identicalresults because it does not make any diference whkh of the otlter functîonsintegrates to 1. Therefore, the whole detnrmînant izt eqn (3.2.10) leads to

(3-2.13)

For readers who fnd it rather dilcult to follow the above argument: 1recommend them to wozk out as aa illustration the case of a second-orderdeterminant, for which

,- - ) j ) ,.:;((,,-,), wwj((,,--), ! ,t- j .,-,,- (('),))wztpz) j gz, (sa , oa.14)(pa(#z)which leads to

G = ) j kwl(p1) 'ztz (n(pz) + (p;(pz) 'Hz wz(p1)) dn ,

aad similarly for Sz. The more general case should then be clearer.

(3.2.1 5)

40 THE IGISENBERG HAVIT.TONIAN

The index i ia eqn (3-2.13) ks that of the argument pq in the integraad.After the htegradon over this mdable, the result clnnot depend on tYsNtkular f. We may as Aell lxke any one of these indices, for tavxmple thefizst one, and wzite

5JJ. *

Q = '-''?a w-k (Jh ) 'Hz w,(/; ) d'p- :.2 %â = l

(3.2.16)

wkic,h Geady shows that the novmn3l-zation factor taken in eqa (3.2.3) ksOrrect. because the reult ks 1 if Nz is replaced by 1. The sum over ï istims

N x

F! & = N% = w*,(#z)>f, wk(#z) dn = s. >i=1 k=k

(3.2.1 Ij

wlzic.h is the energy of these electrons when they are separated from eachother and do not interad. .

Whea the same Mnd of algebra is repeated for khj, it is seem that thecoordinate of iwo electrons are involved in vg = lr# - r#l. Therefore, foreach term of detgtkoé theae are two terms in detlfglz! wkich do not hte-grate to rmo by applying the orthogonality condition of eqn (3.2.5). Thenon-zero terms vzill tims be made out of pklpgj and wv (%) and their com-plex conjugates. The previous argument ahmt repeating the szme integrals(N - 1)! la'rnes applies eqllally well here, and so dœs the conclusion thatthe integraks do not depead on the paeticular choice of the indices ï and j,whiclk may as well be replazed by 1 and 2 . Eence

z*sf'' ,%bt*'l 1 (,2-

'' S'j j ::z. - t j jo k rx p .. bv ', 2 ! w k , ( p r) ) j :) dvj ga2 2 ?*ïjiI.f=Q kxk/uul

v1 - eg- - ' v' (,1 )e-&-' (pa) bs'z.''bt7ns'ok' rpz -' nr''r'.t d'''-2- (3-2.181% ... . s . ''2 rv.fi: 11-' = l

Now, leI l@â(n)I2 is the probability that there is an electron at thecoordinates pï. Therefore, the îrst sum ks the Coulomb intcaction betweeaa pair of electrons, summed over all the pairs. The second sumaatbn ofintegrals cannot be given such a simple classical intarpretadon. It is clear,however, that it somehow comes out of the Coulomb potential because ofthe use of a determinaut, wluich means that it is due to the Pau)i principle.It rnxy thus be regarded as a Mnd of korrectîon' to the classical Coulombinteradion of the frst sumvnxdom wlzic.h does not take Snto accouat thePaali prindple. According to tltîs prindple, two electrons that have the

EXCHANGEINTERACTION

sitrne spin caanot be in the sitrne position, so that their overlap is smallerthan that of classjcal electrore. The ttegrals which appear in the seccndsummation of eqn (3.2.18) are called ezchange ïntlrtzlo. The sum itself iscalled the ezchange enerr term-

lt may be worth noting that the integrals in the enerr terms thus 0b-tained here can be evaluated only if atl the ftmctions' lpkLpzq are known)whic.h is hardly ever the cmse. It is more common to ûnalnate the func-tions vklpï) by minimlcing the total enerr obtained when these enerrterms are substituted in eqn (3.2.9). There are diferent tedmiques for theactual use of this method that mostly rh'1m' by certain simplifbng assump-tions whic.h are introduced before the energy minsml'zation. They are a21known by the general naztte of the Ilartree-Fcok method. Thue VII not bedescribed here.

Consider the second summation in eln (3.2.18), which has jus't beenducribed as the exchaage energy. The important feature of this energy termis that the integzations in it containalsosummation over the spin flznctions-Since these functions aa'e orthogonal to each other, the integral vrill vanishif the spins are not parallel. Therefore, this term actually represents theenergy digosnmoe beiween the state of two parallel spins, aud the state whenthey are antiparallel. W'hen one is interested only in the magnetic propertiesof the material, tMs term may just ms well be replaced by a Hitrniltonianwhich tries to hold the spins parallel (or antiparallel, depending on the signof the appropriate integral) to each other.

.In order to state the substitution of a Hamiltonian more precisely, 1et 11:be the wave functions of the systfem of electrons of the M atoms, when theyare at a very large distauce from each other, nitrnely a certain combinationof the flznctions of a single atom (or ion). Let + be the true eigenfunctionsof the system when the atoms are put closer together, so that they interactmeckly, in such a way that the degenerate enerr levels are split, but notrnz'ved beyond recognition. It is le#timate to assame that it is possible tohave some SOI't of a one-to-one corrœpondence between 9 and % for thiscase- M a substitute forthe true Hamiltonian, '7'f, we would like to have aneffective Hamiltoniam, '/fesj so that its matrix elements with respect to theP's will be the sxrne ms the matrtx elements of the ori#nal Hamiltonianwith respect to the g's? nitrnely ,

('Akl-/ïeel#kr) = ('1z1!7.f1ïIJ'k?)- (3-2.19)Obviously, if the enerr drereace between parallel and antiparallel spinsis going to be the above-mentioned exchange inteval, something that con-tains the sum of terms which are proportional to sL . sy can do the job,where sy are the spin of eac,h electron. However, it is not convenient todeal with ear'h electron separately, and it is better t,o sum fzst over all theelectrons of each atom (or ion) at a lattice point. Some care in -

g out

42 TA'fR HELSEIRBRG HAMRTONJAN

tlsis summation may be ne in some cases, the 6ne details of whjc.hcam be found iu the the treatise of Herring (21), and will not be repeatedhere- The fnal result is what is intùtiyely felt to be the =e, namely tkat

'Ha = - Jé jSï - s.f ,i,#= $.

(3.2-20)

whereze s qZ,J? = 2 k'Jl' (r:)$7;('r2) ,-- ' -- t,7. ? (rg/wyfurt./ dr1 dra . (3.2.21)rà - ral

The conventson is to keep the miuus sign as in eqn (3.2.18), so thata positive Aj mears a ferromMnetic coupling that tends to align spiniparallel to e,arxk other, while a negative Aj means an antiferromagneticcoupliug. It should be noted that in the present Hamiltonlxn Sç . Sy canhave values between -9 and SZ, whereas the appropriate pari in eqn(3.2.18) +4:1,:1% between 0 and LSZ, wllich introduces an e'xtra faztor of 2in the de6nition of Kn (3.2.21). It should also be noted that S: is the t/tllspin of a11 the electrons bound to the atom, or ion, at the lattice site i.For an insulator, the spin is that of a.ll the electrons. For a metal, only theelectrons of the iuner shells are counted, which usually means just the delectrons in the metals Nk Co, and Fe. The conduction electrons of a metalwander around the whole crystal and do not belong to any lattice site, andthe icnermœt electmns are taken as one entity together with the nucleus.

Since the Coulomb lteraction is a scalar? the efective Hnmiltonianmust contain the scalar products of the appropriate spins. Howeverp i.t doenot necessadly mean that eqn (3.2.20) is the only possibility. In some wayit may be regazded as only a Ast-order term in an expansion, the n%tterm of which being

lvI..726 (S,J - S,)2:,.1 )

ét/=land even higher ordersmay be added i'a prindple. As has already been men-tioned in secdon 2.6, the higher-order term is indeed encountered in somephysical situations. Howeverp nothing beyond the îrst term ca!k generallybe included in a quantum-mechanical calculation.

The S'ciscnierp E'amiltordon of eqn (3.2.20) is, thus, the justifcation, ata deeper level, for the a%umption of a force which tries to align neighbour-ing spins. RR>e.U the spi'a operators are replaeed by their eigenvalues, thiRHltetiltoni= 10-* to, and justifes, the &st enera term of eqn (2.2.25). .

Jt is thus the baais for the theoz.y of the Weiss 'molecvlar f eld' apmo-'d-mation that has been ttseê throughout chapter 2, and the bx--K for mostof the rest of this book. In càapter 2 it was also mssumed that only the

EXCHANGB WTBGMLS

nearest-neighbour interaction is usually importaat, and this part can alsobe readily seen ftom eqn rs3.2.21). Since the integral involves the overtapof the wave fanctiop it is quite clear, even without detailed computationsjthat its mlue must decreae vezy rapidly with increui'ag distaace betweenthe ions. In particular, J must be negligible for electrons on farther atoms.Therefore, it is usually sulcient to consider the exchange interaction be-twee.n nearest neighbours only, as has been done in càapter 2-

It may also be added that the treatment here refezred specifcally tothe so-called Idirect' exchange coupling. H many of the fezzites discussedin section 2.5, there is no such direct coupling between the magnetic' ions,e.g. Fe. Instead, there Ls an antiferzomagnetic coupling between the spin ofthe iron and that of aa oxygen ion? and another antiferromagnetic couplingbetween that oxvgen and the spin of another iron in the .same molecule.This coupling, known as a s'upevneltange, still tziu to align the spins ofthose two iron ions parallel to each other, a'ad is egeetiveky the sxme as adirect ferzomagnetic interaction. At the levelof this book, it is not necessaryto distinguish between the two.

The integral in eqn (3.2.21) is symmetric to interchanging t; and j.Therefore, oka = h,iz and it Ls sulcient to take only half the sum of eqn(3.2.20)- This feature allows us to write the Heisenberg Hamiltonian in itsmoze common form, as '

'lfeg = -2 J'l Jç,yS: - S.y . (3.2-22)i>j

3.3 'Rvthaage Htegrnlqlf the fundions yq are orthogonal to each other, adding a tezm with ez/rzar-qn be oected to conkibute a podtive value. This is indeed the c%e forelectrons in the same atom. In an un6lled shelk electrons tend to haveparazel spins as long as that is allowed for the sxme shell: thus cyeating alarge total spin S for the shell. When the fanctions @j are not orthogonal,a rough estimation of the ackange integral J usually leads to a negativevalue. For a problem like the hydrogen molecule, this negative exGange iseasy to understand by a simple physîcal argument: because of the Coulombattraction, the two electrons would prefer to be close to b0th nuclei, whichthey tnn do if they share the same orbit that goes around the tloo nuclei.According to the Pauli prindple: the orbit sharing is posdble only if thespins of the two electrons are antiparallel. Therdore, this antiparallel statehaxs a lower enera than the svte in which the two spins aTe e1. Onecan thus expect the exchaage integral Jçj, for interaction between electronsin diferent atoms, to be generally negatlve. And indeed computations 1omeqn (3.2.21)) for almost any reasonable assumption about the functions wolead to a nmatine excàaage integral.

However, h is known from experiment that Fb, Co aad Ni land some

rare earths) are ferromaaets, and the exchn.nge integrab for them must bepcitive; unh-ke a similar trnmsition metal, e.g. Cuj for which the eigenfuncmtions (p4 are neazly the sarne, but in whic,h that integri must be negative.Tt used to be stated (24) that nobody has been able to compute a positiveexchange integral for Fe. and a negative one for Cu, because rather largepositive and negative contributions subtract to a smaller value that Ls verysensitive to the computational accuracy. More modern computations (22)alleady have the zight Mgn, but the mannitltde of the computed exchangeiategral still difers considerably 9om the experimental value. Improvingthe techniques (23, 24) keps improving the results, but not suëciently yet.The accuracy is certainly not suëdent to afcount for the possibility thatCu may become (25) ferromagnetic under certain conditions.

It is, thus, not possible yet to determine the value of the exdonge inte-gral in the ferromagnetic metals from basic plinciples. One can just assnmcthe Hnxsltonian of eqn (3.2.20), and take Ju as a parameter whose value isobtnsn ed by ftting the theory to a certain experimental value (usually theCuzie temperature). The theoretical situation is clearer in the case of .JerrJ-rztunE-ls, disc'tussed in section 2-5. There, J < % and the basic interaction isantiferromaaetic, but the moments of the two sublattices do not subtractto zero because they are not equal. The net moment is then Kectivelyferromagnetic, in spite of the negative exchange. The theory is also quiteclear for the case of the indirect superexchaage, mentgoned in the previoussection. 1n. those ferrites, one Fe ion is coupled antiferromagnetically to anO itm wiic,h in turn is couplW autifeerromagnetically to Mother Fe ion.The net eFec't ks a ferromagnetic coupling between the two Fe ions, but theiniegrals are b0th negaihe. However, even tn these cases) the values of theexchange integrals have to be taken 1om expeziment, because the theoryis not suëciently well developed to yield reliable values.

THE YLSENBERG HASXTONIAN

3.4 Delocallzed ElectronsThe whole concept of intezaction between electrons wîich az'e loi-mlimzoon ions at lattice sites is at best very much oversimplifed. Afler all, astrong ecchange coupling implies alatge spatial overlap of the electron wavefnnctions, whic.h cannot be reAliz-d if these electrons are strictly loczllezad.Moreover, at least i.a the metals Fe, Co and Ni? conduction electrons aremoving around, and they must also hteract 5.n some way with the electronsat the lattice sites. The picture is cleare.r when two ferromagnetic layersare separated by a non-ferromagnetic metal) and an exchange interactionis carried (26, z7l by ihe conduction electrons of the latter. But even inthe ferromagnet itself, some interaction Ls carried by mobile conductionelectrons- Ja thecse metals, the 3d. ban.d is overlapped ix energy by a muchwider 4s baad, and dnce bands are flled to the Femn'' level) the eledronswhich each atom contribt ;he conduction band are not a2 1om the4s baad, and are partly Le 3d band. Thezefore) the number of d

DELOCALUED ELECTRONS 45

electrons per atom (or f electrons in the case of rare earths) contributingto the bulk magnetization is n:f an integral number, which is indeed anex-perimezt' tal fact .lkhrom the experîmental saturation magnetization of thesemetals) the number of Bohr mn.gaetons per atom is 0.6 for Ni, 1.7 for Co!and 2.2 for Fe. Besidu, the meuured specifc heat at 1ow temperatures iathese materials shows a bigger contribution from the electron gas than r-qn

be possibly accounted for by valence electrons (4s in Fe, Co and Ni)-The theoretical study wkicll is based on the Heisenberg Exmiltonian,

as used throughout the rest of this book, igaores these dllculties, and justputs a nondntegral number ofBohr magnetons at the lattlce sites. There-fore, another theory has been devdoped in parallel, which assume (28)a completely delocallzed: Fee-electron gas, moving in the prcence of thefxed background of the positively càarged ions at the lattice sites. Calcu-lating the actual enerpr b-ds of these electrons can account for the actualspeciâc heat, and can yield theoretical values for the saturation magneti-zation %'s. temperature curves, like the one plotted in Fig. 2.1, az well a.sfor other transport and magaeticproperties in metals. This theory is calledeollnctine elecàzrn ferromagnetism, or itinerant elecfrnn ierromagnttism.

The itinerant electron ferromagnetism is elegant, and some of its resultsare easy to follow cven wîthout detazed computations. Consider for exam-p1e Cu, with 11 electrons per atom. Th%e electrons are suëcient to fII the3d shell, and a fdled shell does not haveany net magnetic moment, becausethere is art equal number of electrons with spin up and with spin down. lnthe 4s shell the exchange interaction is rather low, and the distance betweenneighbouring levels is too large. Therefore, Cu does not have any magneticmoment. In Ni, there are 10 electrons which have to be subdivided betweenthe 3d and 4s shells. In. a gas of free atoms) there are 8 electrons in 3d, and2 electrons in 4s- In a solid, because of the ' of bands up to the same

(Yermy) enerpr leves it can be concluded from the eoerimcntal magneticdata that 9.4 electrons per atom aze in 3d: and 0.6 electrons per atom in4s. ln the unfqled 3d shell the spins are not balanced, because the exchangeinteraction within the atom causes more spins to be up than down. Theexchaage enerr galn is more than sulcient to compensate for the energyloss due to the electrons beingraised to higherlevels in the b=d when theycannot use the lower ones that can only be occupied by those with an op-posite spin to the electrons that are already there. The dxerence betweenthe moments gives risc to a net magnetic moment of 0.6 Bohr magnetonsper atom.

The ma-dmum possibleimbalrce in 3d is when 5 out of the 10 electronsenter thê half b=d with spin up, and the othez 5 split betwen the otherhalf band with spin down aad the 4s. For zL ezecwtrons per atom, out ofwhich z are in 4s and zL - z in 3d) at most 5 rztn be with spin up1 leavingzL - z - 5 in 3d with spin down. The net magaetic moment is then

46

(3.4.23)

Tn Ni, zt = 10 and tr = 0.6, which Ls concluded 9om the experimental valueof lln = O.6/Is, as has been mentioned already- Ignoring the change in theband structure ia alloys of Ni with other metals, and assuming that 0-6electrons per atom st111 go to the 4s band in these alloys, their magneticmoment should be

P,H = (10.6 - njpnt (3-4.24)whicb agrees quite well with experiment for Ni alloys. For example, inalloying Ni with Cu, which haz J1 electrons per atom, the saturation mag-netlzntion decreases more or lYs linerly with încreasing concentratîon ofCuj reanhlmg zero at about 60% Cu, in accordance with this simple relation:

Sa'rnsliA.r estimations for the metal itself) and the efect of some alloying,work well enough for Co. J.n Fe there' are deviations of about 2O% Som thee'xperimental value, which is not surprising because the assumption thatthe enera band structure does not change between one element and theother is oversimplised. But a lineaz relation stizl works for some iron alloysE29), and it Ls ofïen possible to account .for the experlmental results in someothers (30) by very simple models. For Mn this argument breaks dowmContinuing ms before, the magnetic moment of Mn (with n = 7) shouldbe larger than that of Fe; but actually the moment is 0. Pure Mn is notferromagnetic, because the exchange iu Mn is not strong enough to raiseelectrons to higher enerpe levels in the band, leaving the lower enerpe oneswhich become forbidden (by the Pauli exclusion principle) for electronswsth the same spin. After all, eqn (3.4.23) gives the mcmimnm moment,which em'n only be achieved for a stvtmg exchange coupling. However, inalloys with materials such as Al or Bi, the d-istance thetween the 1$4n atomsdecreases sulciently to incremse the exchange integral, and these alloys areferromagnetic.

It should be quite clMr from this outline that the itinerant electron the-ory) with more detGed and more sppkisticated calculations of the enerabands, r-qn be very successfal in interpreting mauy experimental data. Forexitrnple) already computadons which would l:e conddered rather primi-tive today (311 showed, as later confrmed by more elaborate ones (22) 32J,that the enera bands sometimes contnin vezy sharp and narrow peaks.Therefore, a sharp change in some propertie may be encountered when aparticular composition of an alloy passes a certain peak in theenergy baxtd.Also, transport propertia, in pazticula,r the giant mcnetoresistance efect,can be interpreted 133) practically only by the' itiner=t electron pkture)even though lotoliz.xdon does play (34) a certain réle. .

Generally speaking) the itinerant electron theory is quite successful indealing with the whole cr-gstal. However, it Ls quite clear that such a theorycannot hanclle any sppcàicl vaziations in the magnetization, for the simple

THE YLSENBERG HAKTONIAN

y'n = (5 - (n - z - sljos = i10 - (p, - zll#s.

DELOCALIZBD ELECTRONS

Magrglten In kllogauu

47

Flc. 3.1. The magaeœaton distribution ia a llnit cell of bcc Pe, as tseen'bz neutrons. Reprodute *om Fig. 3 of (37) by permission.

reuonthatthe energyband calmzvon is independent of spàce. Therefore,an amitinerant model must msslzme that the magndtiMation is the s=e inthe whole space, while there is strong expezimental evidence to the con-trary, namely that the magnetization in a crystal is a function of spRe. 1.u.partictllar, bulk ferromagnets %ve bœn shown l)y w.ry many t-hm'ques(wbich wfll be discussed in section 4.1) to be subdivide ia* (II-H inwhich the magnetizadon points in derent diruions. 'I'h- eFects, aadthe whole concept of hysterisks as ren jn Fkg. 1.1, must be ignored in a.aatldtinmnt thmry.

Bven besidœ tzhe subdivisîon into domains there is strong expea'men-tal evidence agninKt the itineraat electrozz theory, some of whicil àas beenlite'zvl in a popular review of S--%MK (35j, along with the eezimental evî-dence agnm.' ';:h a tlzeory' that assumes purely localized electrons. She (35, 36jtried to outljne a combined pictme, in wikich m'rt èf the 3d electrons Lslocalized, tàe ot:er part being idneraat. An even clearer picture eAn beseen from Fig. 3 of (371, reproduced as Fig. 3.1 here. lt plots the r-ltz ofshooting neutrons teough an izon crystal. Since tlze neueon bnx a mag-netk moment, it interacts wit.h the moetic Eeld wVe passic throughthe crystal. TH fgttre kterprets the experimemtal neutron data in termqof the magnetic îeld Grough whie.h those neutrons p=. lt is quite obWousfrt)m the flgure that some of the maretizxtion is locoed at the lattkesite, but it is alqn clear that this maaetic moment is very much smearedvound th- sitas. The maaetic Vld distrîbution is complicated, aad the

V

pictuze caanot the even roughly approtmatcd by the naive assumption of apoint maDetic charge at the lattice sites. Some details of this fgure weresomewhat moeed later (381, but they do not change its general propertiu.

There is thus no doubt in anybody's mind that neither the itinerantelectron tàeor.v nor the loc-ql:-med eluron one rAn be considered to be acomplete picture of the physical reality; and that they should both be com-bine into one theorp Such a combined approach may come some kimeybut the present situation is that it hnA not been seriously tried, on atvdunt quaqtivtive levek beause it is just tx diEcult. Workers în mag-netism stick to one theory or the other just because tàey are unable to doaay better. Actually, evo withia eac,h of these approaches there are stilltoo many rhimplifyiog assumptions and apmemations which are ms bad msignoring the other approach, and are acœpted for lack of anything better.H tMs book 1 Goose to use the assumption of laob-vzxd maoetic momentson lattice sitœ, and the Heisenberg Rxmz-ltonl-zm ms derived in sedion 3.2,because it is the only way to include the variation of the magnetization inspMe, whic.h is the main topic of tltis book- It means using a non-integralnnmber of Bohr ma&etons per atom, which is physirltny strange but cmm bezmderstood 1om the forego-mg argttment. Once suc.h a non-integral value isaccepted there is nothhg wrong <th using it in the calculations, and com-.padng the results with experiment. lt n.1qn means ignoring that part of theexchange interaction be'tween the localizM 3d electrons whidz ks cxm'ed bythe conduction electrons. Eoweverj suc.ll a simplifcation is inedtable, aadia any case' expressing tkks contribudon a,l part of the Heiseaber: Hamil-toniaa, eqn (3'.2.20) or (3.2-22), is quite a reaonable approzmation. Aslong ms the Gc%xnge iaYk rlonot be calculated from basic principlesaad it.s value is taken 1om ex-peziment, the experimental value containsthe itinerant eledron conebution anyway.

THE HEISBNBERG HACTONIAN

3.5 Spin WavesThe foregoing should be a'a adequate Jqstifcation for the use of the lnm1-l-tonx-lm

' (z; y ry ;S = -V Jzzzsz - Sz' - V gpzHsg , (3. .z,zz z

where f aœe tke lattice vectors, namely the vetors from the ori#n to thelattice points. This equation is more or less equivalent to mn (2.2.25) whichhRA already been ';> in chapter % only there S were chssical vectors,whereas here they designate the spin operators. The justlcation mxy notbe ms good as may be eetM but it is the bœt we have at tbâs stage of thetheory, which leave a 1ot to be desked. In a way, it is not much more thanan nmmmption, to be adopM 1om now om It is hoped: kowever, that theforegoing argument is suhdently czmvincing for adopting this assumption,

SPr WM'E,S 45

wàich at least is not just an arbitrary, ad hoc azsumption as is the Weisstmolecular feld' used j.u chapte.r 2.

@) and St'/h by the operatorsIt is customazae to replace the operators h z

s+ = s@) + j#F) s- = ,g(=) - jsLN? 4.5.26)z z K : d : : '

for which the Hamiltonian becomes? 1 (z) (a; (z;7t = -.V Jz z. j gxv &-', + h- b''') q + h 5'z, - J-) gpnHh .

.f,,d/ d

(3-5-27)As is known from quantum mechanics, for spins at the aame lattice site,

the original components of the spin S for the same .4 f-.1l'9l the commutadonrelations

j(m) stvlj = ç&,gg(l) ; gsjv) , s'jzlj = jjj,jvjml; r,j.atzl , sz(2lj = jhspt ,: ' : k

(3.5.28)while these components commute for spins at diferent lattice sites, and a1lthese commutation relations mnish for ,4 # .é /. Using these relations; andthe defnitions in eqn (3.5-26),

sçz) s.j- = sjzt gsztxl + jsjvljz z

= jjwl , Asvztrj + sjmt Asvzjph + ,f sjvt ,$vz(,) - .à gs'ztt'l , sjzb ji?t .s'/l + Sjrl sçzt + ,) szç'llt ,$R0 + ?s ,$,z(t)= z z

y, s..v + s+ ,j.(w) a 5 a:;= z z z - ( - -

Hence,

gszto , sz'tj = nékz? h-b, (3.5.30)where é'zz, is the Kronecker symbolj which is 1 if .d = .é ', and O otherwise.Similarlyz

ïfzs S71 = -h%z' ,5-z+. (3-5.31)( z & z

To complete the transformation it is necessary to consider also

(q$k+ , sz'z q = g-s-zt''z ) , .s$ ''') j - i g,s,zC*') , ,sj lz11 + ï / 2) , s,( x'. ) 1 + gs-zt'') : syt'') 1 .

. u k' -' -

(a.-5. -aa)Here the frst andthe last term obvriously vanish, and the two in the midfllecan bc eqraluated by substituting fz'oz!i eqn (3.5.28), leading to

+ S-j = zôzkz, S/) (3.5.33)LS.t , zz -

'50

Actually, the commutation relations are not ttsaally written in tbis fo'rmin books or papers on tbis problem. It Ls castomary to .write them in thespecial (units' in which h, = 1) and therefore omit h. It saves some ink towrite equations in this was but in principle it has a major disadvantageif and when the result Ls to be exprerxsed in terms of 'real units. J.f at theend of the Ypulation the result is a tertasa measurable qnantity' that onewants to compare with eeeriment, it is not always very clear whether thatresult has to be multiplied by >.) or divided by :.2 or whateve'r. lt is always1mach exsier to substitate the particular value >, = 1 if tt is wanted than to,pat in another value if h' = 1 is assumed to begin with. As a matter of hct,theories which use A = 1 (or other non-physical untts, sach as the velocityof light c = 1, etc-) enn ezst only in felds in which theorists compare theirresults with each other's, and the experiment is far removed. It em'n neverhappen in the normal trend of physicsl in which theory and experiment areerpected to go hand in hand. Therefore) it is always a better policy to keepô, in the equations.

Consider now, at each lattice point, aaother operator defned as

@) (3.s.34)N4 = Sh - .Sz ,

THE HEISENBERG RAMK,TONIAN

where S is the spin nnmber of the atom (or) rather, the ion) at that latticetO the eigenvalaes nqh of thesite. Since S5 is the largest eigenwlae of h ,

. *

operator Nz e-xpress the digkrence between the mn.rmum possible value,and the actual value, of the z-component of the spin at the lattice site ,4.Therefore, the nllrnbea's nz are called the spin dedc,titm.& at the lattice pointz. Tuet Tru denote the eigenstate for which the spin deviation is zzo nnmely

Nzèaz = 'rtnli Tru . (3-5-35)In principle this eigenstate is a fuactîon of the spjn coordinates at cJJ thelattice sites, bat such an operator with a pavticulav vatue of ,4 operates onlyon the coordinates which apply to tllis patticular f.

lt is readily verifled that the snmn e kknz is abo an eigenstate of thetX) hose eigenvalae is h(S-zu). Indeed, by using the deânitionoperator Sz , w

of eqn (3.5.34), and substitating from eqn (3.5.35),( z) v .Se Ta, = (SCz - ,5:) tlrvz: = ;;(S - zulllI?w. (3-5-36)

For other properties of this function, consider the expression

sgçz' Jz+ q,. = ( /j# , ,5'zj + ss-bsgçzt ) .Ir a ,.

Substitating for the frst tel'm in the curly bracket fzom eqn (3.5.30), andfor the second term 1om eqn (3.5.36), this relation becomes

SPIN WAVBS

t;r) (,$'G @ ) = )L I'1 + s - vtg j (.çz+ tpazl ;Sz z az s ,

51

(3.5.37)

5+% is also alz eigenstate of Sçtl H fact it is thewhich mp-qns that : w z . )eigenstate which has the eigenvalue h(S - (7u - :t)J. Since the latter is theeigenvalue of kzaz-l, the state Sg-bqçvu must le proportional to 9a.-,. Sim-ilvly, it r-qn be shown that the operator Sz- transforms kbnz to somethingproportional to Taaz.

The behaviour of the operators Sz- arzd &z+ is) thus: similar to that of thecreation and destraciion operators. They create or destroy spin (fetlictbtlns.However, there is a big diference in that the comautation relation of theconventional creation and destruction operators is (a, c*q = 1. If the right-h=d side of eqn (3.5.33) wms a numbevt S/ and Sz- could be normnlizedto make the commutation relaion equal to 1) but that right-hand side isa'a operator and not a number. The best which cztn be done is to defne theoperators

1 + + 1 - scz = S. , ap = Sg : (3-5.38)25 h * - 25 h

for which Kn (3.5.33) become,s

t'Fz z f k' .a. %)hz , tzz'q, 1 = . . k'g ,- ss z (3-5.39)

Nevertheless, it has become customary to use the apmmimation intO the right-han.d side of eqn (3.5.39) is rephcedwhich the operator Sz on

by its eùenvalue Sh. The justifcation is (39J that repbzu-m g that operator byits eigenvalue is correct to a Ez'st ordea and introduces only a secoad-ordererror, at 1ow temperatures. The basic assumption ks that in %he region ofinterest, almost al1 the spins are parallel to z, and the devlations are smallon the average, namely

z'Jz (O) - -%% (T, .n') << M.(0).It should the remarked, however, that there is a diserence between a proof(39J that the neglected term is small at 1ow temperatures, and a çucntitc-tive estimate of how small is small. It is emsy to be coavinced that repladngthe operator by its eigenvalue is a good enough approfmation if the tem-perature is not too high. lt is less e,as'y to say up to what experimentalaccuracy, or up to what temperature, such an approfmation is justiîed.Such a quantétative estimation haa never been done, nor has there everbeen any Auantum-mechauical treatment of the low-temperature re#on byany other approfmation. Therefore, at the present stzage of our knowl-edge, there is no choice but ço accept this asumption because there is noother way to continue the calculation. But it must be bonte in mind thata cea'iain, unspedfed approfmation is involved- Therefore, the statement

52 rl'HE REISBNBBRG EAMILTONIAN

twbich is only too often made) that a qaaatvm-mechanical theory is x'nher-ently more accurate than sopetlzing dadsical (as in chapter 2 here) îs atbest unproved and nncahecked.

Repladng the right-hand side of eqn (3.5.39) by just (%z',

La, , al'l - Jkd' , (3.5.40)the operators c,z and cz' become the same aqi the conventional destruction=d ceation operators, so that

cl k'Ik.n.z = 'n.: + 1 Ww,a-p; , atqn., =, nt Tnr-z . (3.5.41)Moreover, according to the defnition in eqn (3.5.35),

-$% = Jàtzltzz, (3-5-42)so that according to eqn (3.5.34),

60 = $. (S - cl(ul . (3.5.43)Sz.''

Substitutin.g from eqns (3.5.38) and (3.5.43) in ecn (3.5.27),' z'>f = - jy >, ,&z? g.s ((uc1, + (qtuzl + (S - clc,tl (S - (zlzazz ))

z,.f

- JR' ggBS,H (S - cltul . (3.5.44)d

For f # d, the operators commute, and alar may the replaced by czzcl. TJthe names of f and Z are then interchanged in the summation, the secondte'cm in eqn (3.5.44) becomes identical to the frst one. A similar argumentapplies t,o the temnq linear in S when opening the brackets of that equa-tion. The term wbic,h contalns the product of foar az operators is neglected,because according to eqn (3.5.42) it is a, product of two spiu deNiation oper-ators Nz. At low temperatures most of the spins aze azigned, the devia,tionsfrom the fully aligned state are small, and second-order terms are negli-gible. This argument can pnltil y be made (39) more quantitatîve. One e-an

even add (40) the neglected second-order term as a pertérbation, and f ndoui the raage of validity of this approimation. It should be noted, though,that unlike the dropping of the second-orderterm, the approdmation whichhas already been made ih replacing eqn (3.5.39) by eqn (3.5.40) cannot bemade quantitative, and there is not much point iu qu=titzing one withoutthe other. The rcsult is

2&jYJ z, ttzl(uz - G(ul + J'l gIJBD,Ha)aA, (3.5.45)e/g = C - 2zà :L,t .d

SPLN WAVES 53

where2 2 ?

c = -& uh- E zzz? - gpshHslhd,:J

(3.5.46)

and N is the total number of ion sites, namely Ez.The basic assumption is that the ground state for a ferrcmagnet is the

(l)state in which all the spins are aligned abng z. In that state, every Szhas îts max-imum eigenmlue, and there aze no spin deviationd. Therefore,no cz operator can destroy auy deviation ln this state, which is denot. edby To. ln other words, if cz operates on this state, Ta, the rKult Ls zero.And since all the terrns i'a nqn (3.5.45), except for C, have an az on theright-hand sidey

(7Y - Cj gc = 0- (3-5-47)Therefore,

S<7O = C%tO, (3.5.48)whic.h means that tlle ground state 9:) is an eigenstate of 7t) and that Ctas defned in eqn (3.5.46), is the energy of this state.

Spin deviations at any particular lattice plint are not eigenstates of theHxmiltzmian (3.5.45), because a creation at one lattice point, f, is accom-panied by a destruction at another lattice point, âl. Therefore, the excitedstates are not locxlîzed on any one atom. They are made out of spin devi-ations which are propagated throughout the wkole czystal. 11,s descriptionthus calls for a theory which involves the crystal ms a whole, for which oneshould take advantage of the periodic dtr/zclure of crystalline solids. Forthat purpose, the cz operators are expanded in a Fourier series, as is done:41) in the study of the normal mddes of the lattice vibrations, or of anyother property of solids. For N unit cells i.n. the lattice, and one atom perunit cel.l, the Fourier expansion is

a z = vsv 57 a q e ,

-q1 + -yq.z+ - a ett: = q j

q(3-5.49)

where the summation Ls over all the allowed vectors q in the Brillouinzone of the redprocal lattice, quantized according to periodic boundaryconditions. As is the case in any other Fouriear expansion, the inveztedexpansion is

1 -çq.z . 1 . fq-z (a g 5c)(à = (Qe J, = - 5 e . . .q : q L. v z X-N z

From the commutation rdation (3.5.40) it is seen that

.(,.,j = 1- e-ïq-z 1 gcz , czepy e'iq'.t'jq , q x x z,

54 'I'HE MISENYBBRG HAIGTONIAN

1 ï(cj-- cj/).z : ,= y S e = q q t (3.5-51)

because allthe terms in the last sum ' unless q = q/, in which case the

sum is N. Substituting the Fouzier expansion (3.5.49) in the Hamiltonian(3.,$.45), using for the summation izzde,x h = .t - L' instead of L1' and notingthat Jzz' is actually a f-umction of this rekathe dïdtanc: l:l between f and f'and not of ' and d separately, we obtain

1 . -.2q.z y'y a ,yq'.(z-h)'AJ = c - 2A2,$'J-)J(h) - aqe qN:.11 q qz

., 1 ? '

Jl cqzcfq'd + y'l gitzhsïl'- y'l cq* e-ïq'' y'l cq,cïq 'd . (3.5.52)Nqz z q q'

The snlmmxtion over .d in botk terms ks the saae as the last sum in eqn(3.5.51), which is just a delta function. Therefore, after rearraa>g,

-r,>d = C + E 2:2: S J(h) (1 - e-fehl + p/xshlz'j c*qcq . (3.5.53)

q h

Thus 'FJ - C is nearly a set of harmonic osdllators, becausq tzg and ck azereadz- y seen to act as the destmction aud creation operators ln reciprocalspacet The only diference is that the Hamiltoniu of a harmonic osra-lln.toris * aq + .1, whereaz eqn (3.5.53) does not contain the 1a. Eack of theze2o '

atoz's is ckaracterized by a vector q in reciprocal space, but they arenncoupleti to eac,k other, an.d each of them maybe considered independeatlyof the othem. Therefore, the energy levels of e.a,c.11 of the terms in the sumover q of 14 - C are those of a harmonic oscillator without the 1a. Addingthe C term, the eneror levels of 'FJ are

s = c.l- Ezk,q

(3.5.54)

where

Sq = Aqh 2hS V J(h) (1 -. c-fQ'b) + ggjzK p

h(3.5.55)

and sq is a non-ncgativq integral number, which is the eigenoue of (Iq* cq.This nq may be defned as the number of spin 'tpare quantat and the op-erators cq and R dutroy and ceate such spin qmave excitations. Each of

S P LN WhR S 5 5

thce elementary excitatîons is called a magnon. The form of eqn (3.5.54)demonstrates again that C is the enera of the ground state, for which thenumber of maaons, 'rsq, is zero for evezy q.

Now that the energy levels are known, it is possible to construct theJurifàifm l-unction 1om which the physical propertie of a system in thermaleqtzîlibrium can be derived. 1ts general, the stadstical mechanics defnitionis given by eqn (10.14) of (zI24,

z = N-*' c-pe. (3.s.56)V .x 2

wùere p ks defned ltt eqn (1.3.12). The sumrn ation is over all the atlowedquantum states z:, whose energy is &. n'om this ftmction one can obtain,fœ example, the average internal enerr pe.r unit volume,

ë= ksT2 (3.5-57)

an.d the speec heat from its derivative, etc. The average component ofthe magnetic momeut in the direction of the magnetic fe-td is

(3.5.58)

Iu the case of eqn (3.5.54) under study here, the partltion function istims

Z = e-nC J7 f.1 e-nsq (3.5.59)7

Nq

where E% is deâned in eqn (3.5.55).Tàe notation z:q isjust carried ove,r herefrom the foregoing. ActuGy, the summation over each nq is a sllrn overall the non-negative integers, and has notb'lng to do with any particularvalue of q. Therefore, the order of sum and produc in eqn (3.5.59) may bereversed, and the summation may be carried out frst. The latter is a sumof a geomeiric ,serie,s, leadiug to

1.-qcZ = e a . (3.5.60)J& zp&.'s Ej;k J(h) l-u-zq'h x-plzszl'jq v-

1 - e -

Hence

lzfy = &-f) T 1zt Z.- t'?z.r

c -ypuzs Eu J(h) jz-e-iq'hj .pggvyy .1.n Z = - - V Y 1 - e eAT q

(3-5-61)According to the defnition in eqn (3.5.46),

56

(3.5.62)

which is the magnetic moment obtained when all the N spias aie alignedalong the feld direction: z, as is the case at zero temperature. Substitutingeqn (3-5-62) an.d eqn (3.5-67.) in eqn (3.5.58), and carrying out the dfer-entiation,

THE HEJSEAIBERG IIAMDTONEAN

0L-C) r= ggxîtsh = .K,DH

It is possible to continue this igebra in its general form a little further:but at some stage it will be necessary to specify the partîcular symmetryof the crystal under study, and it is somewhat clearer to do it at this stage.Other symmetries can be approached in a si=ilar 'fashionl but the 'wvn.rn p1egivcn here refers specifcally to a body-centred cubic, such a6 Fe, with aninteraction between nearest neighbours only. In this case the summationover h contains only terms for which 1hI = ..4/-3/2 (where .A is the cubeedge), for each of which J(h) is a unicersal constant, J. The atom at (0,0,0)has eight nearest neigkisours, at l.z (+a1.) +a1., +.A), so that

1q .h = -4(+(s + qv + çz). (3.5.64)

Also, this theory' stazted with certaân approvimatîons whîch aJe only justi-fed at low temperltures, and we may as well introduce another one, thatthe main contribution is from long wavelengths, namely small q. The shortwavelengths have a high energy, and it takes high temperatures to e'xdtethem. This argument can be easily made quantitative, because what is usedis a power series expansion, '

h 1 21 - c'Wq' = ïq . h - (fq - h) + . . . , (3.5.65)

and the exNnsion may be carried out (401 to higher-order terms to checkthe eFec't of negleciing them. Eere the series is cut oE for simplicity at thequadratic term. Since the linear term in eqn (3.5.65) obviously sums io zeroin the summation over h, with equal zk terms, the ap>roimatiim we use is

1 .:2'-fq'h x (q . h)2 = - (t/ + q2 + g) (:.5.66)1 - e ,V 8 z M =

plus terms which sum up to 0. Therefore to this approfmation,

S J(h) (1 - e-îQY) gs S .J..42g2/8 = JAt? ,h h

(3-5-67)

SPm WM?:F..S 57

because there are eight neighbours.We also set H = 0, as is customary in this Mnd of calculation. A11

theories of magnetizatîon 'us. temperatu-re deal only sdth the case of zeroapplied Geld, az has already been mentioned in section 2.6 and will befurther discussed in section 4-1. Equation (3.5.63) then becomes

1M a = AG - .ç p s h & s y zt c ç a .y

-cq

(3.5.68)

As is the case in all solid-state calculations, the summation over q may bereplaced by an integral over the Besllouin zone in q-space, provided that theintegrand is multiplied (41) by the density of states, F/(8';r3)? where F isthe volume of the crystal. Howeverv since the exponent in the denominatorcontains (/2, the integrand is ve,ry small for hrge values of )qr,, aud only asmall ezwr rAn be introduced if the intepation is ex-tended over the wholeq-space? instead of just over the Brillouin zone. ln a way tbis argument alsosupplies a farther Justifcation for the approfmation used in terminatingthe sum of eqn (3.5.65) at the quadratic term, because the contribudon ofhighe.r orders in q is rather small. Thus

(3.5.69)

Obviously, the latter approzmation is justiîed if the coecient of /z1)2in the exponent is suëdently large. This coeëdent is about 3 for ironat room temperature, whie,h is already r>ther large for an cponent. Thisvalue implies that replacing the Briilouin zone by the whole space is a goodapprozmation for iron below, and probably up to? room temperature.

The integration over the angles in eqn (3.5.69) is stralghtforward. Hthe integral over q the xadable is replaced by z = Z$VSJAZV. Also? f or abcc the volume can be written as F' = NX2/2, and the number of atomsra%*3 r-q.rl the eliminated by using eqn (3.5.62)

xsfc = J.G(t)) = g;&BItSN, (3.5.70)leading to

s,v awu- . u, - g pjs,;.a j

'' G) q2 sin 0 (Mdp ds2712.$'.M2ç2 . g(0 p &

JG(T) a 1 ks:?* J WA= œ -

g a' z g g y ; g g g ; s z . j .

zG(0) (3-5-715z

Thus, to a frst order at low tempeatures the deviation of the magnetiza-tion from its Nalue at T = 0 is proportional to T3/2 which is kztown as thelBloch law. It its e'xperiment for all known ferromagnets. This Bloch lawhas been derived here only for the particular case of bcc, but the derim-tion is essentially the same (40) for fcc or for simple cubic crystals, and the

58

rœults diser only i)y a numerical factor. A11 these three cubic cases involvethe scae integra:

* vrz gz W=. 3z y

= .-s t.l

-

r tp e -

/3.5.72)t

where (' Ls the Riemaun zeta function. 1.a priadple, the e.xchange integral,Jt cAn be evaluated from the Gperimental value of the coeRcient of T3/2.Howevert this method never yields the same value of J as that which isobtained fzom the ferromagnetic or the paramavetic Curie temperature,as mentioned in chapter 2. The discrepancy is not surprising, because thesemeasurements are done at dferent temperatures, azld there is no reazonto believe that J is independent of the temperature. Even if there is noother esect) theraio eapansion certainly chaages the àisttma between theatoms witil changing temperature. And it is obvious from the theozy thatthe exchange integral, which depends on the overlap of the wave fanctions,must be nery sensitive t,o this distance. It has also been demonstratedexperimentally that b0th J obtained from Tc and J obtained *om thecoesdent of the T3/2 term change considerably when the d''gtnn ces amongatoms are changed'by hydrostatic pressure (43, 44) or by oiher j45j means.Pressure is also known (461 to afect thd h e fteld of the Mössbauereffect.

This theozy of the Srst-order term at low temperatures cAn be (and hasbeen) exetended to Mgher-order terms, as haz been memtioned during theforegoing derivation. H particular, Dyson (401 continued the power seriaGpansion of eqn (3.5.65) and introduced the magnon interaction as a Grst-order perturbatioâ, to check which power of T it Gects. Ei' s result is

(3.5.73)

THE HEBPNBERG HAGTOMAN

with speec expressions given (40) for all these coeëdents aj in all threetypes of cubic crystals. lt is even possible (4$ to remove some of the ap-proyimations of Dyson by the use of Greea f4:. nctions, and obtain whaishould be in prindple a higher accuracy. The disculty is that the expr-sion in eqn (3.5.73) doœ not f.t expezriment. Acmzrate data r-qn be fl.ttedbetter eitàer with an empirical dependence (48j of J in eqn (3.5.71) on T,or w1th a term with T2 before (49) the TS/2 term. The detalâed empiricalexpression for the magnetization of iron whiskers ihat îts the whole range,9om low temperatures and up to the Curie point (50j,

Mz(T) (1 - tj*T)1 = = jjy yyz ,A&(0) 1 - pt + At - Cf (3.5.74)

where t = T/Tc, and p, .,4. and.f are constants, expands to

(3.5.75)

SPDI %'hMBS 59

at 1ow temperatures. h has been suggested (49) that the T2 term originatesfzom contributions of the collective electron fromagnetism, and this ideawas made (511 more quantitative later. Therdore, in this case, as in manyothers? the itinerant and localized electron theories must be combined to-gethez before extending either theory to a high accuracy. Also, measuringMa(T) and ctrapolating it to zero tzrlied jèsld is not always vezy accurate,especially at 1ow temperatures. H some cases the accuracy of the erperi-

al data is not even susdent to go beyond the f rst TV2 term of thementBloch 1&,w, and hi.gher-order terms are mostly of interest to theorists whocompare their results with eprt% other's and not with experiments. ,

For antijerromagnets the situation is much more complicated, becauseeven the ground state ià not as simple and as clear cut as in the case of a fer-romagnet. Approzmations must be introduced already for tlœ calculationof the spin waves at the ground level, and the excitations aze hopelesslycomplicated. There are no conclusions that can be compared with a simpleexpe-riment, or any obvious improvement on the molecular feld. approx-imation presented in chapter 2. Therefore, this whole theoretical f e1d isbeyond the scope of this book.

Other theories which use the Heisenberg Hamztonian of eqn (3.5.25)are not included in this chapter because they either use classical physicsjor at lewst can be o'atlined without specif c mention of quantum-mechanicaltechniquu. One ks the molecular 6eld approlmation, already described inchapter 2. The others will be considered in the next chapter. Howevar,before concluding this discussion of spin waves, there is one importaat con-clusion from the above treatment that xmust be emphasized. The integralin eqn (3.5.69) contxins the factor q2 in the numerator only in three dime' n-sions. ln two dimensions the factor dq would have been qdqdo. In this cxase

(or in one dimension) the integrand with ez - 1 in the denomlnator willdinerge i.!l the vininity of z = 0, namely near q = 0. Therefore any smallperturbation of the ground state will grow rapidly, removing the systemout of the unpe-rturbed state, even at very low temperatures where a21 theapproximations arejustiied and the above calculation is rigorous- H otherwords, jerromagnetic ordering ù not yt).ufslc in one or tvo tfïaendfond-Thjs proof that ferromagnetism (or antiferromagnetism, for that matter)is possible only in three dimensions was already given by Bloch in 1930.It must be noted that it is a fundamèntal propertyj which does not de-pead on any approfmation. The singularity at z = : will be there even ifthe integral is ove: the Brillouin zone and not over the whole space; andthe other approzmations only zequire a su/ciently low temperature. Inpractice ferromagnetism has been observed in some seemingly one. andtwœdimensional systems, to be discussed in section 4.5.

MAGNETIZATION VS. TEMPERATUM

4.1 Magnetic Domn.-nsBefore continuingvth the theories, it seems necKsanr to pa,useand explainwhy the molecular feld approfmation of chapter 2, the spin wave seriesfor 1ow tempeatures of chapter % ms well as cJl other theorie of Ma(T))are restricted to the.case H = 0. For a beginner it must seem naturalto introduce a magnetic feld, at lGst ms a frst-order perturbation) andindeed there is no particular t'he-mtical diëculty in doing so. The cliëcultyis that including a magnetic feld without any other modifcation of theHeisenbea'g Hamiltonian (or of the eaergy of eqn (2.2.25) for the reader whohas skpped chapter 3) doe not have any physical signiâcance, because theresulk of such a Glculation cannot be compazed wiih expem'ment. This factis sometimK forgotten by theorlsts, which makes it even more importantto keep mentioning it.

rlnhe point is that real fezromagnets at zero applied îelds are subdividedinto tbraiz?.s which aze magnetized in diferent directions. J.n other wordslthe directioi of quaatization, z, changes between one domain and another-The rwon for/the efstence of these domains must obviously be a termof the Hazzltonian which has bee,a ncglected so far, but it is too early attlzis stage io specf what this term is, and it will be Rrther discussedin section 6.2. Howeverj the very efstence of these domaâns is a well-established exepeaimental Gactl as has azready been mentioned in secdon1.2. These domains cannot be ignored, because they are being obserce.d byseveral techniques-

The older observytions (52, 531 include ihe Bitter pattern, in whie tinymagnetic pazticles, immersed in a liquid, are attracted to %he high efeld at the ltlalls which sepazate the domaâns, thus revealing the locationof these wazls. ln metallic flms which are thin enough f or high-enerpe elec-trons à.) go through, electrons aze deected by the magnetic feld in thedomains when they pass through them, thus revevng the location of thesedomains. Similarly, the domain structure on the surface clm be seen byelectrons that pass near that surface or that aze refected fzom the surface.Polarized lir/.t reQected from a magnetized suzûce changes its plane of po-larization (Kerr eFect) and its detection reveals 'the dxerent orientationsof the magnetization on the Rndnrte of the various domains. lf the samplels thin enough, the light can pass through it and the iotation of its polar-ization (Faraday efect) shows the direction of magnetizatîon inside these

MA G1N 1!J'1'1G DOM A INS

F '-D .J !' > . *e*' ' ' '. >. ... >* . ... g..-b - 4 '

..> =, . ...i Lw . :' .q :

.' ''..' .' '''''''

.' . lj .t>

1$ .

<

L. .'. r . . . j. . . j': I1 h l .'. . !

. x wœ r ' ='- $ .= ' . G;'. .1 p% .,, fk

1: . . . . ..=' .,m. e%.. a ' 'Nœ+k .=. $.. ' :w1Aw.' '< '. ' A'# 'r ersbnk-q''.'.'f.. p G e ja jz K xj . x ! .,..,'. '< % > II ,- N r. 1 j. N1 ru Ij /t

*r I

.t.' I.1t X' .

. '.. >, g.x'.u.al .wpA. m.z

@1

FIG. 4.1. Three dferent domain conûgurations in the same Ni platelet inzero applîed Eeld, after diferent histohe.s of applying and removhg amagnetic Neld. Reproduced from Fig. 9 of (58) by permission.

these domains. Another method was based on scanYng the magnetic Eeldnear the sample by a very small Hall probe (531 . More recent methodsinclude e.#. passing a current through the sample, and measuring the Hallefect at dxerent pointz (541 , where oppositely magnetized domains #vearz opposite s'ign of the Hall voltage. They also include sœnnsng electxonmicroscopy with polazized electrons (55) and magnetîc force microscopy (56)which allow the study of these domains almost down to atomic size, as wellas scanning optical microscopy (57) which increases the resolutkon of theKerr efect. These, and other, techniques thus leave no doubt that all bulkferromagnets are made out of domains, magnetized in diferent dîrections,untjl a suëciently lazge *eld Ls applied to remove them.

Obviously, the magnetization measured in zero (or .qmaXl applied feldLs an average of its value i.n the dferent domams, and has notbing to dowith tbe theoretical value of Mz(T). Horxver, the measured value is noteven tlnkue, because the domain structure ia zero (or small) applied âeld isnot unique- A good exn.rnple ks shown in Fig. 4.1 which is reproduœd fromFig. 9 of (581. lt shows three completely rlx-Ferent domain confgurationsobserved in the same crystal, after subjecting it to a diferent héstory ofthe applîed *eld. There are m=y other posskbilities which are not shown,and whkch can give zise to a diferent value of the mecured remanentmagnetization. Actually, in zero Eeld it is possible to measure any value of

62

the mavetlro.toa lxwtwpon -Mr and +Mr, as has already bœn mentionedin. secdon 1.1.

Theorists who caïculat.e maaetiza'tion ns. teiperature prefer to ignorethœe domazns, because it ks too dlecult to take them iato account. Theirreasoning is tàat what they calculate in these theoriœ is the ma&eœationinséde tozw.ll of these domm'nq. This quantity ls what would have bten mea-sured if one domain could be separated 1om the others and exïende to''nGnity (59! . The explrimentalist's approach is to measure Mzz for 81Fe,r-eat values of H$ then eMrapolate the (Ia,V down to H = % aad take thatextrapolated value as tEe dtdmstion of Ms. This is tàe proper defnitioaof the spontanec'us 'mcgnetfzcfftrnv Ms, wkich was somewhat ûl-defned insection 1.1. Pzeumably, this exlrapoladon should lead to the value of themagaetization inside each of the domains, as in the theoretiW defmition. .

lt must be emphmsized that the felds applu' ia the exwriment in orderto remove the domains are typically a sma; perturbation, if added as suchto the Heisenberg Hammoni=. For evnmple, ia iroa the îezd necœe.aryto drlve away the domxînR ct vcm tcrnzerct'ure is of tke order of l03 Oe,while the exchange interaction is equivaleat to a feld of 10G Oe. However,tlds raêo does aot nec-atily m/un that tbe eect of the applied feld isjust a third-digit correction) ms it seems at frst sight In the frst place,the applie feld may sometimes be large eaough to crxàe an apprHablemagnetizatioa by distoztiM the atomic electron orbits. Sucb an eeœt is notincluded in the Heisenberg Haailtonian thn.t xq.qnmesfzxed spins at the lat-tice points. But even whe,n tbis efc<wt is negligibleo tke mere arrangementof dornninK kas a ver.g large eGect on the measnzre valuc of the magnetiza-tion, and this process ks not even Enear. Therefore, in rnxny cmses a linearMrapolation of the feld to zero is not Mequatey lpMing to a wAlxtivetylarge uaœrtataty ia tàe experimental value at zero feld. This uncertaintis very often of the orde,r of the dilerence betweea the dxerent theorie.swkich are compared with that experiment, and it coex-nly would not allowaay extrapoh'tion to a ncn-zerc Geld. Near the Curie temperature there isa more reliable extrapoladon teclmique that allows a suEdently lligh ac-cuzacy by a method Hown as the Wvctt plots, whch q'ill 'l)e, described iasecûon 4'.6. Howeverj even there the extrapoladon works satisfactorzy tozero Gelds only, aud there are no eoerimental data for a small feld thatrztn theoretically be added as a perturbation.

It shoild nlnn be '=phasized that atl these theories assume an insnétecrystal. Thcreforc, they caa have a chance of reprœenting the physlcal re-ality only if the size of these domains is mnch 1=#> than tYe cmlationJerwfà, whick is the averMe distaac.e over whick fuctuations of the mag-netization are correlated. Tàis is not always the caa eeecicllp near t,keCurie temperatzzre, whœe the corrdation length diverge.

The maaetization ïnaïde a domain c>n in prindple be measured, inaccordanc.e with the theorehcal deflnstion, by usin.g the M6ssbauer efect.

MAGNETVATION VS. TENJPERA'I'URE

THE LANDAU THEORY 63

In this exppn'ment, a rœonant abxrption of 'prays is obtained when themomentum of tke recril is tTanskrred t.o the wholc crystal iastead of theindividuat nucleus. When either the source or the abxrber is in a maaeticfeld, the enera levels of the nudd are split by the Z-rnan efect, and itLs possible to determ>-ne 1om this splitting the value of the efedive Eeldat the nucleus. In a ferromagnet, this eeecdve ileld tcallu the ltvperjneéezd) at the nudeus is esentiazy proportional to the magnitude pf thema&etization of the atomiq shell around that nucleus. The esect e-an beobserved only in certain isotopœ, one of wikiç:b Ls STFG wllicll is particularlycoavenient for studying fezrcmaaets.

Since this hy e feld is proportional to the magnLtnde of the mag-netizauon of the atom to which this nucleus belongs, and is independentof the dircckrnn of tke atomic moeuMtion, contributioms from the SI'Fenuclei in rlieemnt domains az'e the s=e and Gd np. For the same reaxn,the hyperllne feld in an antife=magnet is the same as Gat of a ferromag-net. Measuremeat of the hypprfne feld as a function of the temperaturethus yields (601 a result proportioni to the maaetization in Fig. 2.1 here,and tlzis meaurememt is indee eArm'ed out in zero applied feld. lt shouldbe noted that, evea though nsclecr energy levels are studied, the nuclearspin dœs not e'nter the calculation of the magnetization: its contribution isnegll#ble compared with that of the s1)1.11 of the electron? leause the Bokmareton is iaversezy proportional to the mass. The nuclear' spln ia thisexperirnent is evntially just a pro% used to memsure the magnetizationdue to the ezccfron spin.

Sirnslar data can be obtained from measuring (61, 62q the nuclear mag-netic rœonance. Eowever, lyoth the M6ssbauer efect cnd the nuclear m' ag-aetic resonance exn at best be uxd for the philosophical desnition of khespontaneous magnetization. The accuracy of Ms(T) mexsm'ed by thesetrenique is rather poor and inadequate. For good quantitative data onemust rely on the measnmment of Mz(.RT) and its extrapolation to H = 0,as in g48) or similar studies. There is thns no way to meuure the magne-tization for H = 0j vith aûy reasonable atcuras'y, and txe calculation ofsuch a quantity has no phyïcal mexnlng, because it caanot possibly becompared with any exDriment.

4.2 The Landau TheoryThe temperature range most popular among theorists is that of the ap-proach tq and the near dfrlnA't.y of, the Cnri. temperatarej Te, becausein thks region it is possible to use frst-ordez approimadols in powers ofF - Tcl. TMs attitade started with the phnnnmenolo#cal Landau theory(63) which applies to all sorts of phase trxndtions of the second knd. ltwill be described here only for the specïc case of the magnelzation goingthrough the Cuzie point- The notauon

(4.2.1)

ks used here for brevity.The basic assumption is that m Ls small near Tc, which mxlrp-q it possible

to expand the thermodynamic potenual per unit volume ms a power serie inm. Neglecting hkgher-order tra.rmq , and auming that there is no maaeticfeld, this expansion ks

*(f, z?z) = ëo(t) + A(t)m2 + #(f)m4. (4.2.2)

MAGNETIZATION VS. TEMPEMTURS

akJa(T) Tm = , t = :Mz (0) Fe

Theoddpowers, m aud m3, are omitted, because ë must remainunckangedby a time reversal, wMch ohnnges the siga of m. The coeEcients #c, ,t4)and B can in principle be functions of other physical properties, whichare ignored here- In particular, it is asumed that everything is done at aconstant yzas.s-?zre so that it is ao1 necesary to specify the dependence onpressure, whick is aa important part of the Hndau theory of othev phasetransitions.

The value of m should be such that the thermodynamic potential isa minimum. A ncessary Gmdition is that the frst derivative vanishes,namdy

#+ a= 2- (.4(z) + 2m S(t)j = 0. (4.2.3)omAlso, in' order for the soluuon of eqn (4.2.3) to be a minlrnum aad no$ amxvivn um, 'the second derivative should be pMtive,

.!. d24 a= A(t) + rvrs S(z) > 0.2 &mn (4.2.4)

The solution of eqn (4.2.3) for the re#on above the Curie point, t > 1, Lsm = 0, and in order for this solution to fulêl the rmuirement ofeq.a (42.4),

.4(t) > 0, (4.2.5)The other solution of eqn (4.2.3), whicN is valid for t < 1, is

X(t)m2 = - , (4.2.6)2B(t)which yields, when substituted in eqn (4.2.4),

:2*= - 4.A(j) > 0.dm2 (4.2.7)

Thereforea noting .q.lpn that the lefwhand side of eqn (4.2.6) is positive,

TEE LANDAU THEORY

A@) < 0, and /(t) > 0, for t < 1.

65

(4.2.8)TMs result means fz'st of all that zrs = 0 is a solution of eqn (4-2.3) alsoin the region t < 1, but there it is a memum and not a nn-nsmnm . Thecombination of mns (4.2-8) and (4.2.5) also means that a coneuity of Aat t = 1 requires that .A(1) = 0. The m'mplest function (and the ûrst-orderapproximatîon f or any o*r ftlndion) whic,h can M6l these conditions isobviously

a1.41) = /41- 1), (z > 0j (4.2.9)where tt is a constant. Fkom this rœult it is already possible (631 to drawsome conclusîons about the entropy, S = -D%f8T, and the speclc heat,Cp = T@S/&T), in which therc tll'rns out to be ujnmp at the Curie poiat.

Let there now be a magnetic feld, E, applie to ïhe - 11 energ.gof interaction with the magnetization is -M . H mr unit xlume, whic.lz is-mSMa(0) in the present notation. With

h = IIMx$4, (4.2.10)the thermodpmmic potential per unit volume becomes

ë(t,m) = ën(z) + @(t - 1)m2 + BLt)m* - :mj a > 0, (4.2.11)where the sign of a is kept fzom the foregoing, for the llmt't lt = 0. By thesxme token, the si> of S(1) should also be kept as in eqn (4.2.8). Thecondition that the Srst derivative wrt-nhes is now

1 . ah = a(ï - 1)m + 2.&(t)m , (4.2.7.2)è'whîch is a cubic algebraic equation for detezmhing gn,.

Above the Curie tempezatute, the righvhand side of eqn (4.2.12) isa moaotonically sncreasing function of m. Thetefore, for eve,o- value of hthere is a single solution of this equation, and it tends to 0 for h -+ 0.For T < Tc, nn.mely t < 1, the frst term ol the right hand side of eqn(4.2.12) is negative, and the equation has three diferent real solutions, m,foz a c-ltin re#on of not too large 1à1 . One of tkese solutions is easilyseen to be a memum and not a Vnimum. Of the other two, one has maatiparallel to h,t and therdoze îts enerr is hecr th= that of 1he solutîonin whkh m has the sam: sign as à,.

According to the dvnition ia eqr. (2-1.22), the initial susceptibilil is

d(Ma) a u. Gr?-xiaîual = lim = (Mz (0)) ,x-n t'9.J2' ,.-.0 ah (4.2.13)

iù the preseat aovtion. By substituthg 9om the deri=tive of etm (4.2.7.2)with respect to h, it becomes

66 MAGNMAXON V5. TEUERATUE.E

E.ka(0))2Mnitzal = 1% '

c -h....e 2 (c(t - 1) + 6S@)m ) (4.2.14)

If h, -+ 0, then m is zero for t > 1, and is given by eqn (4.2.6) for t < 1.Thereforev

v. a . 2a a jf j > 1zstt-z 'Ainivirul =

. a (4.2.15)Jd'a '91 if t < l .41 1-1

TMs di-gence of the snsceptibility for t -+ 0 is the same as in the Curie-Weiss law, dieussed in section 2.4.

The s'pontaneous magneeation for t < 1 at zero fe-ld 1, accordhg toeqns (4.2.6) aud (4.2.9),

?a(1 - t4/ .msp =

) zstj; (4.2.16)

The magnetization induced by the feld, Nud = xH, is in the presentnotation

h pzrïind = X r , xu ér - . . , (4.2.17)CM,COJI Otl - f)

Mcording to eqn (4.2.15). Thae qurtities are of the same order at

(4.2.18)

Hence, a f.eld >, << àt is a tweak; feld, in the ron.v.e that it does not changeGe thermodynamic pmperties of the sâmple. hl a feld h, > hi the therm>dynxmlc properdes have 'valuœ wMc.lz are determined by the ield, and sucha âeld is thus a tstrong' feld. Of course, this criterion ignores the efec't ofthe âeld on the measuzed m by rearran#ng the domes, see sKtion 4.1.It is obvious from eqn (4.2.18) that the trandtîon Eeld, &, vanishœ at thepoiat t = 1) whic.h Ls T = Te. Therefore) at the Cnne' poiat any Eeld is astrong feld Mcording to this demltion.

4-3 . Critizml KxponentsMore modern studies of tàe crzfccl region, nltrndy the near-vicinity of theCurie point, are based on two generi assumptioas, or eoms. The ftrstone is that the asymptotic behaviour govprnsng the approacà to the crkdc,ipoint (ï.e. the Curie temperature) of all physical parameters ls a power1aw in lt - 11, where t is as deâned ia eqn (4.2.1). Strictly specng, tkisstatemeat does not necessatily mpxn that any pazticalar physical quantityis proportional to a certain power of j - 11. lt ozlly meus that it 'aarie.s asthat power, whic.h is more general than proportionniity.

s/2(2c(1 - t)jâe gz z)g -B

XTICAL EXPONENTS 67

The mathemadcal deflnition is that if

(4.3.19)

we Ay Gat flz) varies as P when z tends to zero. Tkis statement iswritten as

.f(z) - zl as z -+ 0. (4.3.20)The simpie.bvt possibility for which eqn (4.3.19) is fulfzlled is when for small

Y Vltv '

lim . = A,z-+o lnz

fLz6) = (lzl (1 + clz + ocl2 + . . .) , (4.3.21)wheze Ct c$, cz, etc-, are constants. However, eqn (4.3.19) is also fulmledin more complicated cases, for dpwmple if for small z,

.J(&) = C I 1.a zl> aàtl + csr'e + ' ' '). (4.3.22)The particular case when the exponent l vaaishes may meaa that .f(z)tuds to a etmstcnt for z -.+ 0, but it may also mean that .fLz4 ct ln z.

In accordance vith this basic assumption, for the l'-m'-t t -+ 1 severaledtfccl eofmenf,s (or crïficc! indiasj are ddned. 1.zt pvticular, for thespeciâc heat: '

Cp .x, 11 - 11-G ' (4.3.23)for the sponlzmeous magnethation below T.,

zrl /x. (1 - t)7, (4.3-24)and for the initial susceptibDity,

Mniual 'xz 1f - 11-t. (4.3.25)General thermodynxmx'c considerations rAn be USGI to prove (632 that ltis the same exponent a for the appronzi of t to 1 from above or frombelow, and similarly fox ''t- Such consideradons also impœe certain relationsbetwen these, and the other, critic,al exponents. nus, for mvample, theinduced magnedzation is

'mind ew hx 'w hl1 - tl-'S, (4-3-26)accordîng to eqn (4.3.25). Using mn (4.3.24)) the feld at wlzicx A1n'R =n.g-netization is of the order of the spontaneous magnetization is

ht '-.z 11 - fI#-F'7. (4.3.27)On the other hand, at this transition betwen a strong aad a wpztk feldthe enerpp of interaction of the feld with the mMneiRation, -:m, shoald

68 'MAGNETIZATION VS. TEMPBRATURE

l)e of the orde,r of the thermal energy., which is of tite order of (1 - ttlcptbecause Cp = -T(:2ë/:T2). Therefore,

ht 'w 11 - àI2-#-G (4.3.28)

Combining this equation with eqn (4.3.27) leads to

a + 2/ + 't = 2. (4.3.29)In principle, the power laws in eqns (4.3.23)-(4.3.25) are based (64)

on experimental observationsj and are not just an arbitrary assumption.The expersmental values are (64) p ;tl 1/3 and 'y ;:er 4/3, whereas the Izaa-dau theors with itxs particularly oveximplifed assumption of eqn (4.2.9),leads to p = 1/2 according to eqn (4.2.16) and e? = l accorêing to eqn(4.2.15). The molecular held approzmation J1.lpn gives # = l/2 and y = 1,as will be clarifed in section 4.6, or as can nlgn l)e seen from eqns (2.2.35)and (2.4.47) respectively. This discrepancy zlustrate the need for a moresophisticated theoretical approach, and indeed thee are more accurate the-ories of these, and of the other, critical exponents in ferromagnetksm as wellas in othe,r critical phenomena nea,r thdr phase transitions. It should benoted, however) that in prindple it ks not dexr whethe a.n eoansion inpowers of T - n should be valîd c Iele away fzom Tc. The value of Teitsdf is deterMned b.r the exchaage integral, J, and the latter may changewith temperature) at least because of the thermal erpansion which variethe atomic distances- If J varies V'I;h temperature, rxl does the tvlnrer)fvalue of Te to wlkicll me==en? at somewhat lower temperatures seemto lead, thus distorting the apparent value of the critical exponents. Alsolin practiO it is not always easy to resolve the leading asymptodc termâom - entxal data, œpecially when the behaviour is of the type ofmn (4-322) here- The nexçorder 'correctiolf to the leading terml a littleaway frt'mv the Curie mint, may be lazge enoul to cunge the apparentvalue of the leading term.

Theorists are never œncerned with these diEculties, saying that themeasarements should be retricted to the vet'y close vidnity of Tc, but thatis often impracdcal. It IUA been noted (651 that the best ft of # = 1 + '//#v4=1% between 4.2 anê 4.7 in the near vidity of Tc. A likely artifad ofhandling the ewerimental data (66) can look like a change in the criticalexponents when Tc is approached. lt is better to use a proper equationof state, over a relatively wide temperature range, as explained in section4.6. It is possible, of coursej to remove most of the experimental data,saying that they are not close enough to Te, but when vez'y f ew points areleft to look at, the data can be îtted to nlmost any value of the critcalexponent. Unfortunately there are indeed some experimentalists who forœtheir data by thîs technique to ft the current theoretical value, so that in

CRITICAL EXPONENTS 69

this partizmlnr feld it is often dilcult to Kay what the experimental resultis. On top of that, the theories always consîder only the tbulk' limit, inwhicE the volnme of tbe system is inqnite. There is nothing fundnmentalia this approach, whch is only a matter of conveniencc, but it must bealways borae in miad that it may distort the asymptotic behaviour ver.yconsiderably. It is aot only that the sample under study must be very largefor sucà a theory to l)e a good approzmation to zeality. It is the size oï eachdomain that must be large enough for such a theory to be accurate enough,aad such a requiremeat îs hardly eveer met. Some of the critkal exponentsmlty be more reliable when they are obtained from the aaalogy with otkercritical phenomena that do not bave the equivalent of a magnetic feld andmagnedc domaias. These are beyond the =pe of the present book, whchdeals only with ferromagnetism aad not with geaeral s'tatistical meckanics.. The secoad basic assamption of the theories of cdtiical cxpoaents isknown as the sœling hplmthadtà. It assnmes flrst of all the estenœ of atorrelabion lengf?l

(' ?x' 1# - 11-M, (4.3.30)

whick measures the average distaace over whic.h fuctuations of the mxg-netization aa'e correlated. It f arther assume that in the critical regioaj thedominating temperature-depeadeace of all the phydcal propertiœ of thesystem is only through their dependeaœ oa this (. If the leag'th scale isincreased by a cez'tain hctoz, the Orzelation length shrinlcs by the same hc-tor. The taperature region, 1-1, then iacremses according to eqn (4.3.30),and a11 the phyical propertie will also chaage by fuxed power laws. Eow-ever, ( -+ x for t -+ 1, axd the lmled system can 1)e rnnomalized, namel.ybe mapped 1:%.,11 on the o

- - oae. T/is procedure is the bmsis of aa

important tool for calmllating the citical exponents) lœowx as the anoz'-mclizction gnmp theory (6:.

Theories of criticae:l expoaen'ks use a general space in d dtmeasions (3in re.al space) aad a sph vector which has zz compoaents (3 in real life).Except for several particnlnmly simple cases, smch am the Ising model inoae aad two dim-nqions (discussed h the aext section, 4.4) or the Landautheot'y etc-, the mathematic ks compEcated. Eowever, it turns out thatthe problem is ver.y much simplifed in the uaphysical coaditions of a verylarge 'rl or d = 4. Therefore, some power seric have been developed whichsbould be a good approfmation for a large n aad a =all

(4.3.31)

These, as well as more accurate methods, have beea reviewed by Fisher (67jaa'd later updated (O) to a certain exteat. They are a11 beyond the scopeof this book.

70

4.4 Isilzg ModelA very popalar method for studying the Edsenberg Hamiltoniaa is a.aapprrMmationwki/ haZ alr-uybeen suggested in the 1925 doctoral thesisof lsing. Tt is baaed on l.eaving out the non-diagonal terms of the spinmatrix, aad keping only t'he components along the Eeld dizection, z.lt means replacing Sz . S2 = Su%% + Sïvsgp + Stxszz by only .%>Saa.And sàce the latter Ommute, it eectively means deRMng <th numbersinstead of matrëcu. lt also meaas that instead of the Exmotom'n.n (3.5.25),or (2.2.25), the crzcrpy of the system is taken to be

MGNETJZATION VS. TEMPERATURE

'.%t-c.zq', - psszzjz ox ,:=-E&F :

(4.4.32)

where every lattice point is iaracterized by a qnxntnm n'tlzn&r cz. I willonly meition in passing that there aze also theoriœ which do the opposite:leave out % and keep only Sz and %. This mssumption, or appremation,ks rnlled the XY-mod.ej aad will not be described here.

In. pzisciple tke Ving model is not a very good approfmation for anytemperature ruge. Eowever, it has the advautage of starting dizectly 1omthe enerpr levels, and sldpping all the steps that lead to them fzom the

.. Hxmotoaiaa, in other methods- This conveaient short-cut mces it possibleto coscentrate on the details of Ge statistical mminMlcs. Therefore, theTm-ng model is very widely used in a vadety of other problems, more thanin ferromunetism for whic.h it waz ori ' y developed. For p-vnmple, in abinnmy alloy mnzlo of atoms A and B, one can defne for tke lattice point .d .the Nalue J'z = + 1 5: there is an atom of the type A there, aad a = - 1 ifthele is an atom of the type B there. If the in%radion is beYeen nearestneighbours ozzly, and SJ 'nxz ks the potential energy betwen two nehbourscf type A aad slml'lxrly for :7z.s and vss, it ts readi)y seen that the energyof any distribution of tàese atoms is #ve.n by eqn (4.4.32), witk

(4.4-33)

but withcut H whic,h has no analogy here. It ks thus possible to studyatheoretiexlly the order-disorder transition (amalogous to the Cuzie or Néeltemperature). At high temperatures there fs a complete disorder, whilebdow the traasidon .A is reglzlarly a neighbour of B for J < 0, and tkereis a separation to re#ons of azmost pure .â. aad almœt pure B for J > 0.Thele is a slight dx-ference 1om the ferromagnetic (or antiferromagnetic)case, in wkiek the direction of the spin r-qJn be rever* at aay atomicdte, while in tNe oxqe of aa alloy, A cnrnot be converted hto B. ln thiscase there is thus the addtional constraint that the totai zwecr of atomsof eRk type must be comsmed. However, the mathematical technique is

1 1 1J = -us - -1)zz - -rss,2 4 4

ISWG MODEL 71

sulciently similar to make it the same nln.m of probl%s. A =iation ofthis case includes the possibility that .4 is aa atom while B is a latticevarxncs whick leads to the theezmodynxmscs of the transition 1om a solidto a kuid or a gM. There are also other physical problems of cooperativephenomena and ph%e traasitions for whicà the Ising model is used, wàicàmakes it belong more in a book on svtistical mecxanics tbltn in one onferromagneem. However, it is historically a paxt of ferromaretism =drxnnot be sldpped altogether. Besida, the problem îas aa easy and elegantsolution: at leas't in one dimension, which is well worth noting.

I wilt restrict the followhg to interactions betwœn nearest neighbouzsonly, and for the caae of spin la, for which duztb of the numbers trz e%n

assume e,ithez' the value +1 or the value -1. There are some more generalstudies in the literature, but they are rather compicated. 1 will also restrictthis section to tke case of a one-dimeuional cbiu'n made out of N spias.The spin at the point t interacts with tke one at &+ 1 and the one at ; - 1)but slce J is mssumed to be the same, the snm of all the interactioas ofspins with the one before them is tEe same ms the sum of Yteractions witEthe spin after them. Periodic boundaur conditions are also &sumed here,namelythat the spia at point N interacts with the one at point 1- Equation(4.4.32) thus becomu for tàis case

N Nzr = -2JE czcz-yz - gIXBH E cz.

;=l 2=:(4.4.34)

This enerr is now substituted in the partition ftmction of Ons (3.5.56)and (1.3.12). For the reade,r who àas sldpped chapter 3, 1 will only remarkthat the partition function is a geaeral stadstici mechanics f cnction, madeout of tke energy levels, 1om wllicà it is possible to derive all the physicalproperties of a system in thermal equilibrbxm. Forthe enera of mn (4.4.34)t/is functioa is

NJfc'ztrz-yz-hzânZ = 8 j

crz=:kl cpvmzi:l z=l(4.4.35)

with the aotatkon2,,/ g;kBHK 'rc r h = - .ksT 2kBT (4-4.36)

Here * is the Boltzmann constant, axkd i1 is hoped tut neither K nor à, isconfused with these letters Msed with diferent mexnings in other sections-For the sake 'of emmetry, 21ö0't is rewzitten as hcz + hcz-h: , because theprodud overtb.e seccmd term is the snme as that ove.r the ftrst one. Equation(4.4.35) is then

MGNETIZATION VS. TEMPERATURE

NA'czc-z-hâez-hhez4.tZ = e .

czztzzbl eN=:k1 d=1(4.4.37)

Themethod used heze toeœuate Z Ls more complex than isessentialforthe one-dimensional problem- Thce is an easier method, but it works onlyif h = 0 is assumed alreuy at this stee, and it cauot be genernlîzed to twodimensions without comple.x aumerical computatioms- The method whkhT am going to descdbe cztn be extende to two (but not three) dt-mensionsalmost without aay change, except that in two dimensions the analyticsolution can only be #ven for h = 0. Jt is actually a onedimensionalformuhuon of the famous Onsageranalyticsolution of the two-dimensionaiLsing model, published in 1944. This soludon is considee to be the realbrealdhrough, and the beginning of all modera statistical mvhxn' '-cs, whichmakes it wort,h studying-

This method simplifes Z b.r deving a 2 x 2 matrix whose elements are

(ezlMlcz') = eK*''''+h*l*h<%' (4.4.38)When J'z and &z, pa through the atlowed Yues +1, the matrh elementskeep going, in a diferent order, through the elements of the szmt matrix,

A'+2A -K: cM = -x g-gud (4.4.39)

where the order ckosen for this pazticular presentation is

( +. +0 +. -. j .Now, according to the rule for mutiplying matriees,

)7l (czlzTIca)(c2IATIcs) = (cz1âJ21ca),trc=uiul

(4.4.40)

)7! )7! (,:IMI&c)(,clMIGs)(GalMIl4) =

<z=+1 <rsci7,

)(2 tc.zlMzlc'altczlM#4 = (o-z1M'1tz.<) ,tra=+l

(4.4.41)

etc. It is possible to introduce the full forpalism of mathemaical induction,but even without doing so it should be quite e-lfxltz by now that when thed-6nltion (4.4.38) Ls substituted in eqn (4.4.324,

ISING MODEL

z = E E F1 (vzlM1a+z) ,trl=t:!:l ex=il z=z

73

(4.4.42)

this matrix multiplication leads to

z = E E f,zIMN-lIcx) lcrxlMlc.:l .

czzzril o'x=éL(4.4.43)

The second index of the last te= is txlctm here a.s 1, becanM for the periodicboundazy conditions it is the same as N + 1 whieà appears in etm (4.4.37).Using again the matrLx multiplicadon rule on the two rpmxim-ng matricœin eqn (4.4.43),

Z = V! @lIMNIcz) = traee (MN). (4.4.44)(r1=;E1

The matrix M i.l eqn (4.4.39) is symmetzic: and rxn be diagonal-lmM.However, for the sake of those rprers who may not be failiar wlth thetraœ of a power of a matzix, let us consider frst a transformation T whic,hdiagonnlî'zm a geneal matrh, M, namely

-z lz 0T MT = - (4.4.45)0 Aa

Multiplying both sides of the equadon on the right by T-IMT leads to

c lzT-LM T = O0 j z-1 up . ( j

' à0 j

2 m ( j ,tj j y (4.4.46)12 2

whicx cam obviously be generrdx'qed to kigher powers. Since the traze doesnot change by such a trxnqformatiozb eqn (4.4.44) can be written p.s

Z = AW + X* (4.4.47)1 ')

wherq 1: ald lz are the dgenvalues of the matrh deGned in eqn (4.4.39).It can be safely asspxmed that this theory is only used fœ vezy large

values of N, and even nearly equal numbers become very dierent whenraised to a Wge power N. nswfore, if .:: > la , the second term in eqn(4.4.47) is negeble compared with the &st one, as long as there is nocomplete degceracy. It is thus snmcient to take

Z = kNz . (4.4.48)Fb'xgonn.llming a 2 x 2 matzix am eely be cvried out analytirlmy. Aecord-ing to eqn (4.4.39) the equation to be solved is

MAGSETIZATION VS. TEAEMTUM

eN+2& - A e-K

! e-K cA-27z - à ) = 0, (4.4.49)

whicNlA2 - 2AeX cœh(2:) + 2 sinhtzr) = 0. (4.4.50)

This quarlratic equation for .à %nK two solutions, tke larger of whic,h is theone with a + si& ia fzont of tke square root. Substituthg in eqn (4.4.48),

Nz = 6K cos%(2à) + ezK sinhzoi,' + (j-2-ri-

q )

Tke magnethation is #vem by etm (3.5.58),

M. = EBTu lnz

9IXBN tî c.q. izaztah,l + o-ax= yé ln eff cos:(2h) + c s2 .

gIXVNeK sinll(2h,)= -. . (4.4.5z)ekK sinb.2(2h,) + e-2XV

(4.4.51)

Obviously, this magnetization vaziskes for h = 0, so that the system isnot ferromagnetic. However, it is tnlmost' ferromagneticl in the sense thatthe magnetizatîon t=d all other physical propeies wkich may be derivedfzom Zt are extremely sensitive to magnedc felds, even when these Neldsare quite small- This feature is s-n in the sqaare root in eqn (4.4-52)1or already in eqn (4.4.51). Tâe 6m+ term of tMs square root vanishes forà. = 0: but already at ratker small values of h it bexmes bigger 1%n.n thesecond term. Tke reason is tkat accordhg to eqn (4.4.36), K is small oïyat Mgh temperatres, while at 1- temperatures K > 1j which makesêK > e-2X. Thezefore, at 6m-te but small applied îeld the second termia the sqaare root ks negligible, the hyperbolic sine fcaacels' betw-n thennrnerator and the denominator of mn (4.4.52), and the magnetizationlooH'as if it e>apolates to a 6n5t.e *ue at zero feld. lt may be eazie.r tosee this efec't in the initial susceptiblty,

f'lMz . a ax -ax -a/zxukitiaz = lim ct 1!Tn (4h e + c ) t (4.4.53)H....z t'?.?2' x....()

after dropping higher powers of h. It is a constant fœ h = 0, but for non-zeroh, the xcond term become negeble, and it seems that tke susceptibilitydivergœ ms h-n3.

ISING MODEL 75

The average internal enerr of the electrons, per unit volume, as givenby eqn (3.5.57), is in this case

(4.4.54)

in zero applied feld. TMs energy contributc to the specifc heat of a unitvolume of the crystal

z- kszc 0 uz - .-z-wtanh Lk2J ) ,#/ sT

0ë 2.z 2 c.zcn = gg = knN ( srl sechz jazj , (4.4.55)

which is a continuo'as function of the temperature. Studies which startfrom the assumption H = 0 (as lsing ori#nally did) use tids result asan indirect proof that the one-dimensional lsing model predicts no stableferromagnetism at any temperature. The point ks that in a transition fromorder to disorder the magnetic enerr (especially the uchange energy)must be converted into something, so that it takes an extra heating at thetransition, which mupt appea.r as a jump in the specsBc heat. Indeed) eventhe simplest theoriœ (such as the Landau theozy in section 4.2) predicta discontinuity of the speclc heat at 7lj and tMs jump ks observed inall experimental evaluations of the specifc heat of the electrons-Yquation(4.4.55) gives a rounded peak, but not a discontinuity.

Here this conclusion was reached from a more elegant calculation of theactual magnetization and initial susceptibility, in eqns (4.4.52) and (4.4.53).Moreover, this method showed that the study of one dimension is not com-pletely academic, because the system is talmost' ferromagnetic? which mustmean that some small perturbations may make it a real fezromagnet, aswill be discussed in section 4.5. This method can also be extended withno particular complication to cover the Ising model in two dimensions. Forzero applied f eld, that problem has an analytic solution not only for asquare lattice: but even foz a rectangular or a hexagonal one (6$. Detailswill not be given here) but the result is that the Ising model does give stableferromagnetism in two dimensions.

H prindple this result is wrong, because the true Heisenberg En.nn1'1-toniau cnnnot support ferromagnetism in less than three dimensions, asproved at the end of section 3.5. Howeverj it only takes a rather smallmodifcation to have real systems which are jerromagnetic because theyare 'nearly' one- or two-dimensional) as will be discussed in the next sec-tion. For these cases, the Ising model is a very usefttl theoretical description,and indeed its results are in reasonably good agreement with experiment.It is not as accurate as the more sophisticated theorio, but it applies tothe whole temperature range in one analytic solution, which makes it aconvenient tool to use.

76 MAGNETIZATION V$. TEMPEMTURE

In three dimensions, or for a spin larger than 1, the Ising model can2only be solved by applyiag further approximations, or by using complicatedmathematical techniques and computations, or b0th (69!. For these casesthe Ising model has no particular advantage over other techniques.

4.3 Low DimensioniityStrictly spealdng, neithez ferromagnetism nor antiferromagnetism can efstin one or two dimensions, at least ia as much as the Heîsenberg Hamilto-nian, with a11 the study around it, is a good approvimation to physicalreality. The proof of this statement wms given at the end of section 3.5,and the reader who has skipped chapter 3 should just taake my word for itthat such a rigorous mathematical proof exists and that it is undeniable,involviag no approfmation. There is also a dxerent proof (70) which isbmsed on another approach, but leads to the same result, nn.mely that nospontaneous magnetization (or sublattice magnetization) can exist for theHeisenberg Vnml'ltonian in one or two dimensions. However, it was shownin the previous section that the lsing model for one dimension is tnearly'ferromagnetic; i.tt the sense explained there, so tàat even a small pertur-bation may make it a real ferromagnet. Therefore, a system which is only'nearlyt a one-dimensional Ising system may well be ferromagnetic (or an-

tiferropagnetic), as long as it is not sirictly one-dimensionat. One suchsystem can be a set of one-dimensional chains, with a strong exchange in-teraction within each of the chins, and with a weak exchange interactionamong the chains. This additional interactiol can be slllcient, in somecases; to sta'bilize the ferromagnetism, while being too small to afect theresults of the one-dimensional calculations.

Suck systems do est i.n zeality. The one moétly studied is the crystalwhich is made of moleculœ of (CHz)4NMnC)a, also known as TMMC, fortetramethyl Jmmnni'um manganene chloride. ln this material the chains areseparate' d by about 9 â, so that the interchnx'n coupling is (71) at leastthree orders of magnitude smaller than the intrRhain (antiferromagnetic)exckange interaction. For such materials, the theory of one-dimensionalchains is indeed a good approximatëon (71, 72j to the experimental ruults,including the memsurements (73J of the susceptiblty and of the speclchcat of the electrons (after subtracting the contribution of the lattice). Ofcourse, the theory is not necessarilyjust the lsing model and more complextheories (74, 75) have also been developed.

ln two dimensions, the lsing model does #ve a stable ferromagnetismwith a well-defned Curie temperature. A simple physica: explaaation ofwhy ferromagnetism cannot eOst in one dimension, but can n='Kt ia two,ca,n be seen on pages 309-10 of the book (41! by Ziman, but it cannotchaage the lct that the more general Heisenberg éamiltoniaa does notallow feromagnetism in two dimensions. Obviously, the ïsing mssumptionthat the of-diagonzl elements of the spins are' negligible is a sufdent

LOW DIMRNKONALITY 77

perturbation tllat can stabilize the ferromlgnetism, in the sxme wa,y asoth6r pcturbations ca.n Sometimes do it. In Kme way 1he Ising model,which allows interaction in the z-direction but not along the w or the wdirectionsj is a particular case of an anksotmvic ezc&mge, which is notthe rsame as what is describe in sedion 2.6 under the sxme nxme. In thepreent context it mpm.ns that the Gchange smteradion is of the form

(m) ''-=) (sr) s (s' ) s ( z ) s '' z )Jx S.2 Sj -1- Jg S,q f + Ja j i,h ,

with unequal &, Jv, and Jz, and this assumption is sometimes used evenin theories about crystals with a high smmetrp It has been noted (7OJthat such an exchange may be suEdent to stabilize ferromaretism intwo dimension? However, this kird of an anisokopic Gchange e.xis-ks onlyin some theoretical studiu, and there is no experimental evidence for itspossible etstence. Among other perturbations whick may also do the sp.me,it was shoqm (76) that with the efstence of a dipolar inteaution (whoserange is insnktelt a twodimensional system may become fmwmcnetic.A magnetic feld may also stabilize ferromaaetism, or at least make thesystem Iook Jik: a stable ferromagnet. .'nus, the initial susceptibility ofa tw-dimensional system rxn obey (M a power law, and diverge ahwea certain transition tmmperature, even though it doœ not really becomean fordinay' ferromagnet below that temperature. The same was lotedby Mermin and Wagner (70J, who remarked that their proof rules out only .spontaneous magnetszation, but it 'does not exdud.e the pomibility of otherb'nds of phase transitions', such as a divergiag initial susceptibility belowa certaân temperature. '

Besides all those cases? the ferromaretism may be stabilized by a smalliateraction between hyers. As is tEe case in one dimension, there are alsocrystals which are salmost' two-dimensional systas, because they are mMeof hyel's with a stzonguchangeinteractîon betw-nthe ions in them, wVethe interaction beimeen the layem is much weaker. In such crystals (711 73i

. the twœdimensional theory âts the experimental results quite well.Morecve, the emperimental study of magnetism in two dimensions is not

restricted any more to materials which occur in nature. Shce the inventkonof molecular be-xm epitaxy (AmE), a whole new class of artifdalstractureshas beea made and studied. These are superthin, clean single-crystal flms(even down to one atomic layer) separated by a11 sorts of non-magnetic hy-ers of any desired thickness. These hyers are built up (78) into very regularsuperstructures which allow detailed cxpen-mental (794 and theoretical (80Jstudy of both three-dimensional stmzcinre, and ialmost' two-dimensionalones. It Ls a whole new world, which allows the detaâled studies of efec%tllat have just been neglected or wrongly evaluated some years ago, sucxai the propertie of the snrfaces or the exchange interadions cnmeed bythe conduction electrons of a non-magnetîc metallic layer between mag-

78

netic layers, etc. In particulaz, it is cleaar now that a venr small iateradionbetwœa uitrathin layers can make the whole stucture ferromagnedc (orantiferromaaetic) but strictly two-dimenskonal monohyers are paramag-nets.

Kstoriely, the problem of magnetization in two dimemsions was ap-proached by the study of thia 62msj cvaporated in vacuam, wllich was nota very hi@ vammm in those days. Htead of tàe many Jayers with weakinteraction between thvm, as in the more rv>nt study meationed in theforeming, tme Glm was mnze out of atomic layers 5n coasad, namely witha RI-,/ strong exckaage interaction betw-xt tàem. The quesdon whicx w.xsvery much distmsaed aad argued wms kow thick suck a layer must h beforeit has tbe magaetic propertjes of the bulk material-

Until 1964 the spin wave theory predicted a considezable reduction inthe spontaneous maaetization of îron alr/uz!y at 1û0 i, or even at a largerthekness. Caltulations udng the molecuhz 6eld appremation gave largerMs down to a much smallcu.r thîckncs, but nobody took them mrsously,because tkce classical rcults mast be much poorer than the quaqtum-mechecal ones. Experimentally, tke decreaseof Ms with decremsing thick-n- wms evem fmster thaa the spin wave calculation, aad theorists madeeforts to modifjr aad corred their calcalations i:a that direction. The frstGception wu the case (8IJ of4lmq madein a betlervacuum tllan everybodyekse's, whio gave xdrnos''t the bulk ma&etization in Ni flms made of onlya few atomic layers. This result ftted the molecnlxm âeld approzmationywhicà a16.0 prokibîts magneeation at one atomic layer, but predic'ts Mswhicb ks only afew % below the bulk *ue at tmo a'tomic layeri Obviously,this experlment was ignozed, as were Overal which followed.

The real break-through was a zero-feld Mössbauer efect ' ent(82), wkicb el-vml'nxted the problem oî reaching saturatioû for very àhîn flmsand the possîblty of magnetizatlon created by the appliH magnetic feld.Eighly enricxed (92%) WFe was usedl but even +at ms not suRdent formemsuring a single layer. Theefore, many layers were madeo separated bySiO, which iatroduced some uncertainty in d ' their average thickness.Still, the results were clear and showed a very great discepaacywith a largenumbe,r of previous '

ents: for axt iron thickness of 7.5 â. the Curietemperature is 83.,$% of its bulk value. The roomrt=perature h nefeld is only 4% below its bulk value at 6 k thlckness, and drops to zeroonly at an average Shickness of 4.6 A. Tkue results are quite close to thepredichon of the oversimplised molemklar feld approzmation.

Thks experiment raised maay hea*d diseneons aad arguments. Theo-rists Gopted, (83) the new rontts mther laicldy, and the thKretical spiawave calculations soon ftted them. Eximliztàe:lltnll'em took longer to be con-eced that all tkeir previous results weze wrong) and kept arguins (84) thatthe thickness of the new (82) flms was not memsured properly, or t%not some-thing else wms wrong thexe. They were only coneced after the Gxperiment

MAGNETIZATJON VS. TEMPERATURE

LOW DIMENSIONAJJW 79

(85) in which a ssngle Fe layer was used aqi a sourte, instead of an absorber.This flm was measured in the same mcuum chamber i.!l whc-h it had beeamade, without ever exposing it to the atmosphere, tkus avoiding ofdation.

. Later mod-l4cadons (86) nr.t:.d a much higher vacuum, and studied tke e-Q-Hoî a slow deposition rate, or of stoppiag the deposition for a wbile andthen conthuing it, or of heating the substrate, etc. Tàe conclusion fzom allthese studie was that there is no ferromaaetism in the limit of one atomiclayer, but that it take only a little more thickness than that to stabiWethe ferrom&gxetism- R turned out that the 61=q in the older expersmentswere not continuous Fe flms, because they weœe heavily Ydized. To avoidoidation, Alrns must lx made rather quickly in a mlldently hizh vacuamand then dther kept in t'he vacuum or covered by a protadive laye.r Morebeing exposed to alr. nus the SiO used (82) for separation turned out tolx aISO a protection against Odation- If the fzlzms cre allowM to ofdize,tkey become sexrate islands of Fezczfclds rathe.r thau a Gmtnuous layer.Titus the magnetintion loss eve,n for rathe.r œck fllms vas (8% 87J due tothe sexradon iato isohted islands, and noi the eAect of fllm thinlcnv. Thepoînt is that small enough ferromaaedc pxrticles aào lose thdr magneti-zation by an efec't knou as snpeoaumagnziLvn which will be discassedilz section 5.2.

A11 theories and experMents thus point to the absen& of ferromag-netism in two da-mensions, unless it is stablzed by one of the 'ways men-donM in the foregoing. There ks, howeverj one possible exception. Thetompearatuze-depmzdenc,e of Ge electron spln rconance of Mn2+ ions wasmeasured (88) ia a tlitcrally two-dimensional' layer of Mn atoms, made bya cerfxin Gemical proœss. At about 2 1(4 the resonance 6e1d decreasedabruptly (by more than 103 Oe within 0.2 K) in a maaner which is typicalof a phase kansidon into the menk yerxzrlcgnetidzrl mentioned in section2.6. As mendone in the foregoing, not eveuthing whick looks Dke a mag-netic is one, aad t'he evideace would have bee,n more convincingwith a difea'ent measurement, instead of the spin resonance which involvea large mavetic feld. However, s%e magnedsm with a veU 1ow Cm'ieor N&1 temperature is not really ruled out by the foregoing arguments. lfthis experiment proves to indicate a real Mtiferromagnetism, it only m<orsthat the Heiseaberg Hamiltonian is aot the G1ll stors aad some Gditioniterms skould be added. The possîbîlit.y of dipole intazactions, buîdœ theexckaage interaction, has already been mentioned in this sedion, and aNée,l point of 2 K may well be possible for it. After all, most experimentare not carried down to very Iow temperature, so that t'his pazticulaz re-sult may not be unique. Besides, tke whole concept of a two-dimensionallattice may not be very accurate down to atomic siza, because the spin isnot a point charge. Each îon also has a three-dimensional structure, evenif it is not necessarily as pronounced as in the case of metallic iron shownin Fig. 3.1.

80 MAGNHTIZATION VS. TEMPBMTUEE

4.6 Arrott PlotsThe interpolation techniqtte known mq the Arrott plots was frst snggestedorally by Arrott in a conference with no published pro-dings, then dis-cuRsed (togethez with othe methods used at the time) in aa unpublished(89) internal report of the General Electric Co. It is based on a powe,r seriese-xpxnm-on of the BrGlouin function, (2.1.15), whose argument is small inthe vicinity of the Curie temperature. The be ' '

g of this exp=sion hasalready been given in eqn (2.1.20), but here it is carried to one more termin the expansion of the coth lnction, yielding

s + 1 2s2 + 2s + 1 s gtzsj .Ss(z) = as j -

xsa :r + (4.6.56)

Subsdtuting eqn (4.6.56) in the molecttlxr feld basîc formula (2.2.33) andrdumraaging the terms, we obtain

3s z;j g + 2S2 + 2S + 1 ah = gs + 1

- a( acsc (?& + G#) ' (4.6.57)

Nmn.r the Curie point, the initîal susceptibility divergœ, wbich means thatV/z is small- Therefore, powers of h, hgher than the fr-rt are neglected.Dividing eqn (4.6.57) by >, the ldt hand side should vanish at T = Te,which means that

35a(Tc) = . (4.6.58).% + 1

Hence,H 2- = a(T - Tc) + bTM. , (4.6.59)kh

where a aad è are constants. It is not diëcult to write these constantsexplidtly, in tez'ms of the pltumeters of the molecular 5.e1d theom butthat is not nec-- The important point to be noted is that they are notfunctions of H, Mz or T, and depend only on the type of ferromagnetîcmaterial

The frst conclusion from eqn (4.6.59) ks that for H = 0,

-?G2 x (Q - T) -1 t (4.6.60)

and thatXioîtsal cx Mz/A o: tT'c - T)-Z . (4.6.61)

According to the de6niûons in secticn 4.3, this means that the criticalecponents for t'he molecular âeld approomations are /3 = 1/2 and ..f = 1,az stated without proof in that section.

The second conclusîon fz'om that muation is that if expezimental datafor Mz at diFerent felds and temperature are plotted as Mzz p.s. HIM.. at

AKR OTT PLOTS SI

const=t tempuatures, they should be straight liues in the kriiical regiont:namely when temperatures are not very faz fzom the Curie pott. Theintercept of thue Dnes with the (HjMz)->m's is posîtive if T > Tc, andnegative if T < Te- The advantage of this ldnd of plotting for an accuraiedetermination of Te is very dear and obvious. It should only be noted thaithe data for 1ow îelds, ihat do nd ft these stzaightlines, must be discarded,because they represent avezaging over domna-nn whicx are magnetized itzd'lerent direciions. This point was Alws-qzly empha-zed in the GE report(894, which waraed that in these equatons Mz 'repreents the mrvquredmagnetization of the bulk materials only if domain alignment is complete',which means avoiding âelds that are too small.

Even when thee plots are not siraight lines (because real matersalsdo not obey the molecular feld thez'y) they are still qaite useful (902 fordetmmiaing the Curie poht, because of the clear distinction of the iatvceptfor the t=peratnre to be ahwe or below Tc. However, there are decultiœin Hrapolating mzrves, because the human eye cxn only really deal wîthstraight lines. There are n.lpn diKculties (90) in dedding where ihe Emitof the low-fcld data 1. Therefore, it was found Gtter (91J to include theproper critical ezptmezlt.v of section 4.3, and tzy to ft cll the experimentaldata în ihe critkal re#on to the eqnation o.f state,

(4.6.62)

1/pwhere the parxmetezs 'y and p are chosen so that a plot of Mz 'p:.(S/MaIVT at a constant T gives a 'set of staigh.t lines. This set of plots,whick was given the nxrne tArrott plots') becaae the standard techniqueused by rnn.ny workers as routine. E8wever, three points which werc em-phuized in that paper (91) were hter forgotten or ignored, and are worihrewating here:

1. Equation (4.6.62) is only one of m=y po%ibilities to lteep the sxrneexponents p and 'y at the critical region, and the choice may dependon how wide this region is dp6ned to be.

2. The values of the exponenis p and 't cannot be deternn-nM âom theft io the experimenial daia io any decent Mcuracy, becaase the ftloolcs vezy much the skltme over a wide range of the values of ihœepxrxmeters-

3. There is no way to ph-mlnxte the curvature of the daia for nery lomJdds. The bœt way is to ignore them; see in particular the datapoints in Fig. 3 of that paper, and also (65).

An equatîon of state has also b- proposed (50) for the whole rangeof temperatures, not only the critical regiow and is #ve,n by mn (3.5.74)here. For t $:, 1 it becomes the same as eqn (4.6.62), but only for H = 0-

ï/@ z - z y.r 1/#H c . z= +sfa Q Ml

82

An empirical genervzation of eqn (3.5.74) which should apply to any Eeld(gom zero up to saturation) has been proposed (921, and compared withsome experimental data from the literature. Eowever, thae data did not goup to a Mgh feld, and the agreement was not really any bettez than thatwhich could be obtained from eqn (4.6.62). Also, the ft was not very goodlmainly because c2î the experimental data were used for the least-squareftting, whereas the low-feld data should have been kept out of it, becausethey represent only the rearrangement of domains. At any rate, this ideadid not catch on, and nobody else tried to use that equation for any otherexperimental data. It should be noted, though, that both Fig. 1.2 and Fig.2.1 ia this book have actually been plotte with the use of that fozmulaland with J = 0.368 and c = 1.112.

lt was frst noted by Wohlfarth (93) that the Arrott plots should becomecurved, if the material is not homogeneous. Eis approach was ex-tended(94j by several others, and later used (95, 96! to explaân some featuru ofthese plots in amoIpho'as fezwromcgnets which are very heterogeneous in-deed. Nevertheless? this theory was forgotten., and for severa.l ye'az's thesame curvature was attributed to some special properties of amorphousmaterials, predicted by a theozy based on a ,fzrst-order perturbation of theEeisenbergEamiltonian, whichwas supposed to apply at nery low mcgnetï:Selds. In spite of a1l the wnma'ngs aginst such an approach, as emphasizedizb.the present section and in section 4.1, dozens of experimentalists hur-ried to produce low-feld Arrott plots, to compare with that non-physicaltheory. Details are beyond the scope of this book, but it should be notedthat by introdudng the amorphidty as a Gaussian distribution of exchangeinteractions, an excellent ît to some experimental data was obtahed (97)provided that tlte ltv-/eltf data 'tpez'e eîclnded Fom the Stting. This theoryused 8 adjustable parameters, not a2 of which were really necessary, andthere waa actually no dcculty in keeping, for example, the accepted the-oretical mlues of the critical exmnents p and 't, which would have give,nalmost as g*d a ft. It was just a tactical error to Gsist on showing, inthe same paper, that the experimental values of these exponents are un-reliable, by ftting the data with very d'Xerent values of # and q. H thisfeld of critical exponents, theorists got used to telling the experimentaliststhe dcorrect' xalues to which they should ft their data, which is againstthe tradition of physics ia any other feld. Therefore, showing that the datacould be ftted very well, f or example, with y = 2.2 was taken as a heresy,and neither the Pltysical Sede'tp nor the Jonmal @J Applod PFzrt/àc.s wouldpublish it. lt was eventually published (97j in Jotzmcl H Magnetism crdMagnetic Materials and ignored by everybody.

MAGNETIZATION VS. TEMPBRATUE.E

5

ANISOTROPY AND TIME EFFECTS

5.1 Am*rsotropy .

The Heisenberg Hxmîltoniaa is completely isotropic, a'ad its eaera levelsdo not depend on the direcdon in spaœ i.x wlzicx the eryst.al is maoetized.Throughout the previous chapters the measured magnetizadon was consis-tently denoted by Mx, where the z

' 'on is the dizecdon of tb.e appliedfeld. lt do% not reily have any meaniag in the lirnl't of zero appëM feldzfor whic,h most of the œculadons have b-n done. In fad, the concleonfrom a2 the calculadons described so far is that a fcromagnetic hasa certain magnetic moment Jzj whose z-cmponent is a certain function ofthe temperature. We How that at 1ow temperatm'œ mœt of tb.e spH areparallel to z, but this z has not been defned yet, aad will be introducedhere. .

Eoweverj before de6nlng tYs direztion, it ks Kustrative to consider thebehaviotu of aferromaaetic Nrdcle in the case of complete isotropy, whena11 direcdons in space are equivaleat and the choice of z is crsitmrp. At1ow temperatures, strong GxcAange forces hold the spins parallel to eachother, and the dizection of thee spins defnes the Oection in space of themagnetic moment >, wkc.ll is g)&B times the vectorial sum of the spins.Let this Jz be at azt aagle 0 to a %ed magneic seld H. The enerr ofthe ineRtion betw- the feld azd the magnetizadon of the partide isHown to be -lts'= 0. TheMorey at thermal equLbrium tàe probabilityof having a particular =gle 0 at a temperakzre T is proportional to e OS 8

whaeyH

z == , (5.1.1)NTand ks is the Boltzmann constant. Heace, the average for aa ensemble ofparticles is

J2'*' J= eos 9 eœcoe# sh 9 dp d4 (tcosp - 1.) ezcthedl %'

(cos p) = -.0 0 = x u (h = z,(z),zx 'r mcosp su o ts ds lea'xloepl'''Jo lz e û(5.1.2)

where1

f?(z) = e-oth z - - (5.1-3)is exlled the Langeein fmction. It is readily seen that the Langevia functionis the limit of the Brillouin lnction of eqn (2.1.15) for S -+ x.

84 ANISOTROPY AND TIME EFFECTS

The left-hand side of eqn (5.1.2) is the component parallel to H of aunit vector i.a the direction of the magnetimation, by the desnition of theangle 9, np.rnely

MH >N== (cos 8) c= 2) , (5.1.4)IMI ksT

which proves that a2l ferromaoets are actually just paramagnets. Andthere is no mistake in this algebra: there are only two diferences betweenthis calculation and the study of a gas of paramagnetic atoms in section2.1. One is that the function 8 is continuous here, while this mriable haddiscrete values in section 2.1, and the other is that the magnetic momentJz, was that of a single atom there, while here it is the moment of a largenumber of atoms, couple'd together. However, thc second dilerence is onlyquantitative and not qualitative, and the frst one should not make anydrerence, especially since the energy levels of a large spin number S arevery close together and look like a continuous mriable. lt is thus tmte thatif there were no other enera term besides the isotropic Heisenberg Hamil-tonian, it would have been impossible to measure any magnetism in zeroapplied S.eld, and there would be no meaning to a Curie temperature? orcritiY exponents, or aay of the other nice features mentioned àn the previ-ous chapters. Theorists who calculate these propertie never pay attentionto the fact that the possibility of measuring that which they calculate isonly due to an extra enera term, which they always leave out.

Of coursez a magnetization as in eqn (5.1.4), which is zero in zero ap-plied feld, conlradicts not only experiments that produce Fig. 3..1. It is alsoin confict with the everyday experience that, for example, the particles inan audio or video tape stay magnetized and do not lose the recorded infor-mation when the writing feld is switched o1. lt is because real magneticmaterials are not isotropic and not a11 ulues of the angle 0 are equallyprobable. There are several types of anisotropy) the most common of whichis the magnstocrystalline anisotropy, caused by the spin-orbit interaction.The electron orbits are linked to the crystallographic structure, and bytheir interaction with the spins they make the latter Jzre/'er to align alongwell-def ned crystallographic aaes. There are therefore directions in spacein which it is easier to magnetize a given crystal than in other directions.The diference can be expressed as a direction-dependent energy term.

The magnetocrystalline enera is usuazly small compared with the ex-change energp The magnitnde of Ma(T) is determined almost only by theexchange, as in the calculations of the previous chapters, and the contribu-tion of the anisotropy is negligible f or almost a11 the known ferromagneticmaterials. But the direction of the magnetization is determined only by thisanisotropy, because the exchange is indiFerent to the direction in space.Therefore, the axis z of the quantiMtion direction is always a directionfor which the anisotropy enera is a minimum. lt has nothing to do with

MISOTROPY 85

the direction of the feld H, even if some of the phrasing in the previouschap%rs may have led to the conclttdon that H is always pazallel t/o z. lnreal Dfe the Eeld rnxy be applied at any angle to the intnvnal direction ofthe anisotropy a'Hs, as has be% hiuted at h Fig. 2.2.

It may be worth noting that theoriu efst for the case of a largeaOotropy energy, which is aot negli#ble compared with the exGange.Lf such materials could be found, their magnethation (981 and eve,n theirCurie point (99) would be dxerent when measured ia diferent diredons. Itshould also be noted that adding anisotropy is not suëdent yet for subdt-viding the (xystals into the domaiu mentioned ln section 4.1. The exchangetzies to alir a11 the spins paeallel to each other, and the anisotropy triesto align them along a certain crystallographic direction. Together, they tr.gto align all spins paratlel to that directien: and the diviâion Y'tO domainsmust be caused by ret another epergy term) to be discussed in seuon 6.2.However) once the domazns are there, the anisotropy energy term will tryto align the ma>etization in e-ac.h of them aiong one of the axes of its en-err minimum. The domains are thus regularly arrangd along well-deznedirections) and are not randomly oriented, as Weiss ori -

y assnmed.Quantitative eeuation of the spin-orbit inteacdon from basic prind-

ples ks (100J possiblel but the accuracy is iaadœmate, as is the case withtke exchange integzals- Therefore, anisotropy energies are always wriiten asphenomenological Gpressions, whicà are actually power series expansionsthat take into account the crystal syznmetrs and tke coeldents are iakenfzom Gperiment. Specifc ex-prfmlo:as e>n only be writte,a for a specifccrystalline smmetur, as Ls done in the following.

5.1.1 Uniazizl XnïsofromThe anisotropy of haagonal crwb-lal.s is a funciion of only one parameter)the angle 6 betweea the o-nvn'q and the dsrection of the magnetization. lt isknown from experiment that the eaera Ls symmetric with respect' to thecyplane, so that odd powers of coso may be eh'mlnated fzom a power seriesexpansion foz t:e anisotropy enerr dens,. Its frst two tarms alwe thus

= -.% cos2 9 + Kg c>4 $ = -Jqm2 + Kgmh (5.1.5)Ou z z)

where z is parallel to the czystallographic c-axis, and m is a unit vectorparalle.l to the magnetization vector)

Mm = - (5.1.6)IM1

The subscript 1ù is nrv!w.d here, because this Hnd of anisotropy is usaallyreferred to as a nniax one. The coeEcients Kk and Kg are constantswhich depend on the temperature. Tkeirvalue are taken from - ents.H principle the expansion in eqn (5.1.5) may be carried to Mgher orders,

but none of the Hown ferromaNeiic matRn-nh seem to require it. H most=es even the term with Jfa is negligible, and mxny experiments may l)eanalysed by using the frst term only. And ia all Howncases IJGI << IXtl,whicà Justles the power series expnnm-nn-

Mœt workers prefer to rewrite mn (5.1.5) as

tu = A'l sit2 I + A'a stnt # = .&:(1 - m2) + A%' rj: - rrt2 hy2 (5.1.7)J: z 1

ia which case the coeRdent Jfz has a diferent tlclve thaa in the case ofeqn (5.1.5), nnlejs Jfc = 0, or is ntr/kibly small. Once Xz is pro-lyrèdgmmed, the dâference bemeen eqn (5.1.5) aad mn (5.1.7) is a œnstanitand a constant energy term does not have any physical mpxnin g: it onlymeans a shift in:the defln-ttion of the xo energy, whicx ks neve importantfor the problems disfmme in this book. Therefore, the choicc between eqn(5.1.5) aad mn (5.1.7) is complelly arbitrary, ms long as the defnition Lsnot switched in the middle of a calcuhtion. tn either case, both Kï and Jfamxy l)e either positive or negative- In most hexagonal crystals, the c-nM'q

is au 6asy cds, which meaas it Ls aa energyrn-,ninz xm aud not a mxvlmum.Ja these cases, Kz > 0 in eqn (5.1.5) or >xqn (s-1.7). Were are, htlwever,mater-nlq for which Kz < % aad for theem the c-ax-k ks a hard Hs, w1:,1z aû

easy Jlcne perpendiculaz to it. Some hexagonpl ferrites have AlKn a ctArLni'nxmount of auisotropy uâtldn (8) the ayplane, but it Ls iways smaz, and Lsat mcs't just barely measrable. It will be ignored here.

ANBOTROPY AND TM EFFECTS

5.1.2 Xbic Xnis/lrop#For c'tzsïc ccstaà the expansion shoid l)e uachaaged if z Ls replaced byp, etc., when the axes z, y, and z are desned along t:e czystallographicaxes. Again, Vd powers are nlled out aud the lowœt-order combl'nniionwhich fts t;e cubic symmetnr Ls m2z +m2v +m2z) but tllis is just a constant.Therefore, the expamsion starts Gt.N the fourth power and is actually

= X (m2m2 +. a20,2 + 0,2.2 ) + Kgv,z .2.2 (5.1.8)'J;c 1 x y p z z z z y z :

where here the values of JG aad Jfa are also taken from experiments, andthey also depend on the temperature. Here again the expaasion may becarried to higher order: but it is not necœsary for any knowa ferromaaet.

' k rs prefe to replace the exprerxsion with X1 by -'

1(zr/ + m4 +Some wor e z . vzr4l, but witàout changtg the second term with K1. This substau 'hondoes not nloange the cpeEcients, because

4+m.*+m.4+2(p12zr:2+,r:2,r:2+,r:2p:2) = (m2+w2+w2)2 = k. (5.1.$mx t!l a x v v z a x = v z

Cubic materials ezst with either sign fœ Kï. For nvxmple, Kz > 0 for Fe,so that the easy axes are along (100), while Kï < 0 for Ni and the easyaxes are along the body dîagonals, (111).

MTISOTROPY 87

If M is the same everywhere, the above expresions for the eaergydensity have to be multiplied by the volume of the c to obtain theaaisotropy enerw. Ho=ver, if M (or m) is a function of space, as Ls thecase in some problems discussed in the followiag fin.pters, the eneror is

S = J m dr, (5.1.1 0)

whee 'tp stands for either 'tt/u or 'tt/c (ox any othe foz.m of anisotzopy, ms thecase may be) and the integration is over the volume of the ferromMnet.

5.1.3 MzgnetostrictionThere are o' ther forms of anisotropy beidu the magaetocrystalline one.0ne of them is due to an Kect whie had already b- observed in the19th centurs aad #vea the name magnetosyctiow when a ferromagnet ismagnethedy it shn'nv (or eq3a11ds) in the directioa of the magnetization.Strictly spduBlng, sac.h an ect inml-vdate even the deînition of M as thedipole moment per unit volume, because the Cunit volume' itself changœwith the magnetizzation, which changes wîth the applied feld. lt Ls alsoquite clear that when domains are maretized t=d therdore change theirdsmensions) in dxerent directions, there caa be a misft of the crystnllînelattice at the boundary between suck domn.inq, wbicx would lèad to aztutra strain energy. Such eec'ts have b- studied (101, 102) for somesimple cases, but the problem of the mavethation in a delornvble bodyis outside the scope of this book. Even i'ts mathemntical formuladon (103)Ls ex-tremely complicated aad ha,s neve.r b-n Rllly developed; not even forthe case when the sample Ls (104) mMnetkally sa*rated. lt is thereforecsumed here that all bodie are Bid, and all these magnetoelaztic efecfswill be jus't oored. Only three remarks must be made before droppingtMs subject.

One Ls that a large part of the energy of the internal magnetostriction ina ferromoedc czystal rAn be expressed in the same mathematical form asthe uniafal or cubic magnetocrystalline anisotropz gîven in the fœegoing.When the coeEdents A'1 and Kg are calculated from basic principles, thecontribution of this magnetostriction should be added. But when the OeE-dents are taken fzom eoeriment, this contribudon is alrevy induded, andnothing is rpmlly Vglected by not mentioning it. The second point is that itis possible to add another dimension by mGsuring fezromanetic crystalsunder pressure. To a îrst-order approvl-mation, suc.h - ents e%n bexnnlyM 21G5, 1G6a 107) as = extra aaisotrop.y 1fvm. ()f cou-, the eas'yaxes of this t-rm depend on the applied pressure, and do not necessadlycoiodde with the crystallogmphic =e,S of the material. In these casu taadalso in Hme c s with only au internal straân and no external pressure)it mxy be prBmïble to have *th. cubic and unieal anisotropy terms in the

88

sxrne nmple. The third remark is that some ftrst-order theoriœ e-fst fœthe e;u of internal straias at custalline imperfections (in particular aœrtain distzibution of dislocadons (108, 109) or impurity atoms (110) andother ddects (111J) on the approaG to saturation. There are also many (.:x-pœiments which show (112, 113, 114) that tlïc introduction of dislocations(by m''lBngl and their removal (by aanealing) ha.s a hrge efect on the mea-sured coercinity. This eeed is connected with the large magnetostriction inthe vicinity of the dislocations, for which there Ls a detailed theory g115),but it is outside the scope of this book.

AMSOTROPY Ar TIME EFFECTS

5.1.4 Oler Cae.sOther forms of anisotropy include the shane cnfsofm.py, originaung frommagnetostatic properties, whicx will be dinntu%ed in section 6.1. t'11 thecase of thx'n magnetic flms there is also another form, krttvn ms indcedcrlïrezv. It was a very popular sublect in the 195œ and 1960s, whenthere were maay expersoental investigations of thia ftlms, mostly madeof permuoy, which is an alloy of about 80% Ni and 20% Fe. It was thenfolmd that when the Alm is deposited at an oblique angle to the substrate,or when a large magndic (or even electric (H6J) feld Ls applle dlm-ng thedeposëtion, a uniafal azdsotropy of the fnr'm of ecm (5.1.5) or eqn (5.1.7)developed in the plaae of the âlm. Applying and removing a maaeticfeld (with or without Rmmealing the sample) could also induce a uniatalaaisotropy that was usually referred to as a =tatabêe xnl-Kotropy. The latterwaa also' observed in the bulk (11C and in cobaltlll8). Eowever, in spiteof the wide Tinterest at the time, the orîgin of these phenomena has neverbeen Gxlly established. The conduoa' of a 1962 review (119) was that theinduced xnl-sotzopy is very compEcated and it is not fully understood',aud in 1964 the p

- was (120) that the problem tis too complex for

a complete quxntivdve eeatment', wVe a 1969 paper (121) stated thattthe meclmnl-Km - - - is a mzbject to be iavestigated'. The mcuum used inthœe (1a.> was not gxd enough, see section 4-5, and it is qutte po%iblcthat cxygen playe a role (122) ia some of thue efeds. lnhomogeaeities ofrcomposition (123) aad of phase (124) were shown to be part of i% and thepfwqx-ble eed of impuritiœ was demonstrated (125) by the enhancement ofthis nniKokopy when uother meual was codeposited with the permalloy. Aniaternnl strxi'n mny have-also (126) played a part. None of these Gects wa,sever fully clarifed, nor was there any real admnce lalr, and some featuresof the induced anisotropy are not easy to t.:xpln.''n even in the more modeneoeriments. It was not so much tkat the pzoblem was too diEcult, butthat most people jast lost interest in thue knds of experiments, althoughsome are still (127, 12% 129) being reported.

An interesting feature of the permalloy 6lms was that they were poly-crystalline, ï-c. they were made ofr small crystals whose custallographicaxes were randomly oriented. Therefore, they had a local cubic anksotropy

ANSOTROPY 89

whose easy axes were also randomiy oriented, besides the ovmrxll unia'dalanisotropy. TMs random anisotropy caused (130, 131) a rfmle :tacjuratEe direction of the magnetization ia each of the domxins wiRled mghilyaround its average direction. The theory of this ripple (or wiggling) wasquite straightforward (132): and was A.1M v-n-Bed 7331 by ferromaaeticreson=ce. The exchange and ovea'a,z anisotropy tend to keep the magneti-zation in ezmh domain parallel to the Mm!A.='x1 easy aais: while the randomauisotropy tria to tilt it into a diFerent Oection for each crystxllite. Thecompetitbn between them rœults in the former two strong forces. keep-ing the magnetization direction nearly constant, but they yield a little tothe weakerj random term, allowing a small tilt in each crystallite towardsthe locd easy A=-K of the cubic anisotropy. nen n.mozphons fcrromloetsweze ârst made, it was still taken for granted that the same argumentabout random anisotropy in the small permalloy crystnlllte appliœ Just mswell to the random local anisotropy of the ions. Therdore, the eFect shouldbe similar g134), namely there should be a ripple structure with a (Mrixa'nsmearing (95) of the critical region. The non-phykcal theory (critidzed al-reMy towards the end of section 4.6), according to whicx the occurrenceof a random =isokopy, no matter à/'tc smallt must lead to a drasticallydxerent qnalitatkne bGaviour, cngne only later.

When a certain think'ness of the Glm is deposited with an anisotropyinduced along a chosen di-ction and the rest of it is deposited with thexn-lsotropy iniuced alonga d@erent direction, a spedal form of a sïczïcl or.even tyexdcl q13,$) anisotropy is obtplned. Other spenlnl, artiidal type.s ofanisotropy have been obtained by depvtion (136) on a scratched substrate.

5.1.5 Snöace WzzfaoàmzpThere are several contributions to this term, the most importazd of wltdcàwms suggested bac,k in 1954 by Néel, who poino out the importance cfthe reduced symmetzy at the sudace of a ferromagnet. The spin at thesuore IUA a neazwt neighbour on one side and none on the other side,so that the exnbn.nge energy there cannot be the same aa in the bullc. Anon-mavetic metal depœited on a ferromagnetic one #ves (13*6 138, 139)an evem deerent eavironment for the surlce spins, and so does (140q theintmrlœbetw-n two d@erent ferromagnets. The easiest cue to considerijthat of a thin flm, because in this case it is possible to compute the actualwave f unctions for eacà atomic layer to a reasonable accuracp Calcalationsfor a few atomic layeers aze possible and show (80, 141, 142) that it LG notonly the efect of the lat laye on the surfazeo but it mesrz'ia inwards to afew more. The problem is more complicated for other geometries, and itis not even clear to what e'xetent the results on thin Glms are applicable tothem. However, from a phenomelfolo#cal point of view, any surface enerrterm should be a tendency of the surface spins to be dther paraoel orperpendicnlnm to the surface, ia the rtme way as the thin Alrn enerr term is

90

all esotropy whœe easy nan'n Ls (143, lz14! dther pazuel or perpendicularto the Slrn plane. Therefore, to a Ot-order approvimation any thezyshould lead to an enera term of the form

AMSOTROPY MVD TWE EFFECTS

1 cs. = yft'a (u - m) #,5, (5.1.11)

where m is defned in eqn (5.1.6), the iatepation is over the surface of 'tàeferromaoet, and n is a unlt vector parallel to the normal minting out ofthe surface. The coexde'nt Ks should lx *e.n from experiment, but thereare not many dear-cut experiments which mraluate this parameter, and itsvalue for any give.n ferromavehc material is often contzoversial.

The form of eqn (5.1.11) mssumes 214x% that the sndarp esoàopy Lsa geometrical featme that depends only on the sltape of the surface. lt isalso possible to imagine (1461 a surface anisotropy caused by the reduceds'ymmetzy 'of the spin-orblt interaction at the surfce. lt can lead to an

enerr that depods oa tke angle betwœn the magnetization at the surfMeaad the awtaaographic axe,s of the material, beside, or instead ot ew(5.1.11). Computations 1om 'Iaalic pzindplps on single-crystal flms (142)contain both jossibilities together. They could be designed to show theGCSC't of each one separately, and the quutiox could also be clxrif M bypropely designed experiments (791, which has not been done yet.

The e'nergy term in mn (5.1.11) is the frst indication in this book of apossible spaa-dependence of the magnetizadon. If the surfRe anisotropyprefers a rlleerent direction from tkat of the zlnllkntzrlyy in the bulk, it isconcdvable that the ma&etization vector wi!l poht along the bulk easyn='q in most of the crystal and will the.11 gradually turn into a diferentdirection when it approaches the sarfaze- Of courseo lt exn happen onlyif the surfMe Ysotropy enea'g.r is large enough to compensate for theworkthat needs to be done agm'nqt the exchaage energy, which prders fullalignment. It is illumimating to think of ftA''K posslbilit.y eve.n at this stagebecause it contains some of the importut features of the magnetœtaticOergy that wm be htxduced in secîon 6.1. These dxerent cases nl= sharethe conamon property that thcsy are automatknlly iaozed ia a calcalationthat mssumes azk insnite crystal, which does not have a surface.

5.1.6 'Eqmvimental MefzodsThere are several methods for measnring the coeEdents Kï and Ka of themagnetocrystxllln e anisotropy. Usually the total xn-lsotzopy is measured,and the shape anisotropy of the swple must be known and subtracted.The most common method is known aa the iorque ctzzva the c'rystal ismagnetized by a feld applied at dceyre'nt angles to the crystallovaphicaxa, aad a tomion bnlxnce is used to measure the resultiag mechanicaltorque. 'ne applied feld must l)e (1411 large enough to remove the magnetic

ANVOTROPY 91

domaîns (148j but not so large that it afHs (149) the memsured valuœ.Even au eledru feld may somdimes afct (1501 the m=ured valuœ-The' mathemadcal form of the angular dependence of the torque is usuallyknowa as one of the etpwt--ons in the foregoing, or a tfansformation (127)of them, but the analysis of the data is also possible (151) in some caesfor which the symmetrz Ls not known in advance. TV method js usuallyapplied to single crystals only, but uader certain condidonsj torque cuzvœcaa (113, 152) determine the distribution of tbe magnitudc of aaisoeopyin a powder wit: racdom dlections of easy afs. Measudng 61mn Ythdlferent thieaesses rxn also yidd (153, 154, 155) the snöace Ysotropy.

Other methods have to rely more heavily on the theoretical interpretotion of what is measared. They include;

1. Measurement of the magaetization in large applied felds, f.e. in whatLs Wova as the approach to .scftlmïftzrz re#on. 1n this re#on it issuëcient to use a linear theory by neglecting higher orders (1561 ofthe magnetlation component perpendicular to the applied îeld, andthere are n.l!m (15% empirical rules.

2. FerromMnetic resonaace ln the geometzy of thin Glms. The theoryis well understood (see section 10-1), and the analyds of the dataeAn yield not only the bulk xnlgntropy constant, bat also/that of thesnrface (143, 154, 15% 159, 160! aisotropy. '

3. The transnerse initiat suxeptibility, dvned as

8Mn= Em -

,Xz agJJx..+0 z(5.1.12)

is plotte versus a béa Jcîd HA. O1d calculatitns (161J were based ona certatn model of Stoner and Wo%lfa.rth, wûich will be dîscussed insection 5.4. This model n.%cumes that the sample is made of particles,and that there is no spacedependence of tke magnetization witàineach paztide. The o1d theory predided cwnçs in this susœptibilitywhen H. reaches oae of the value -KïIMs and +2KïlM>- Suchcusps could noi be seen g1611 in the older experiment's. They werelaër found to exist (162, 163) (although i.n the form of rounded pealqsinstead of c'usps) in fILCUgVaiIIGI ferrites, but not in coarse-grainedones, whicb must be subdivided into domains. This technique Ls quitepopula.r nowadays, especially (1641 foz matlrinlq witk large valuc ofKz, for which it has been descrîbed (16S) as 'euy to hxndle' andûan interesting alternadve' to other methods- Improvements in thismethod (166) allow the evaluation of Kg as well.

4. Singularitiœ in the zamlles susceptibility were also predicteê (161)but not observed, till a more complete analysis (167) 1ed to the meaesurement of the derizabine of the suscepdbility, namely &*tMxl0H2z.

92 ANISOTROPY AND TIME EFFECTS

This method works (168) for coarse grains, when each czystallîte con-tains domains. If this deri=tive ks detected by its second harmonicresponse, the distribution of anisotropies (169) cmm be measured.

5.2 SuperparnmagnetismBefore introducing amother enerpr term, it should be instructive to studythe change which the introduction of the anisotropy made in the calculationat the beginning of the previous section. Consider a group of particles, sa.yspheres for example, hadng a uniaxGal anisotropy as in eqn (5.1.7). TA Kzbe nesected for simplicit'y, although including it does not rfolly complicatethc calculation. It only require to choose a specifc value of K2(K1 for anyparticular evxm ple. If the magnetic moment p. of a particle is at an anglc 8to the easy afs z, and a magnetic feld H is applied along z, i.& at 0 = %the total energy is

S = XiF sin2 $ - gKcosot (5.2.13)where J'r is the volume of the particle. This function is plotted ia Fig. 5.1 vs.#. Obviously, the Boltzmann distribution cannot be used as in eqn (5.1.2),because not all amgles are equally probable a zzvforb. There are two mirdma,one at 8 = O and one at 8 = Jzïy whose enerbes are

à7l = -gH alld (5,2.14)respectively, with an energy barrier between them. In thermal equilibrium,the magnetization will tend to be in the vkinity of these minima.

Actually, for such a confguration the question is not what thc thermalequihbrium is, but whether that equilibrium ls reached at a11 under normalconditions. As a rough approzmation one can assume that the magnetiza-tion vectors of the particles spend cIl their time in one of the dircctions ofthe mimsma,.and no time at a2l at an.y direction in between. ((n. that case, thenumber of particle jumping over the barrier from Mnimum 1 to minimum2 Ls a function only of the height of the enerpr barrier, Sm - Xg where Smis the energy at the memum (see Fig. 5.1). The lattcr can be evaluatedby equating to O the derivative of eqn (5.2.13):

sin 8 (2A%V' cos 8 + pHj = 0. (5.2.15)The solution sinp = 0 leads to the two minima whose en'ergie are givenby eqn (5.2.14). Thc other solution js ihe mazmum, at

yHcOS8 = - .2N17 (5-2-16)

When it is substituted in eqn (5.2.13), the cnergy at the maximum is foundto be

SUPERPARAMAGNBTISM 93

Jm 7

Rz7

I 1 0JJ zz vrA

FIG. 5.1. The ftmaion of eqn (5.2.13) plotted for p,s' = 0.15JGF.

2+ HM 2p. szk = Aezy + = Aez&r 1 + .4.Y:7 2A% (5.2.17)

The second relation is obtained 9om the deinition of g as the magneticmoment of ev.h of the particlœ, aad of tke magnetization vector M as themagnetic mcment per uait volume. n1K defnition mpxns that g = FM,namely g = Msï'r, since M, as defned in sedion 4.1 is the ma>tude ofM in the absence cf magnetic domains.

Therefore, the number of particlesjumping over the barzier 1om rnlm--mum 1 tc minimum 2 per unit Mme rxn be written ms

-pfrm-4'àl = cste-plkhvjk+Hllçx? (5 a )8)n2 = f1ze , . .

where c1a is a constaat, # is defned in eqn (1.3.12)j and

2A15r 2TGFx = . = .

p. Ms (5.2.19)

Simjlarly, the number of particles jumping over the barder from rninianm2 k) minimum 1 per unit time is

-#(:m-8a) . .-#xyv(1-A/Ax)2 (5 z zc)L'2l = t;21e Az , - .

whe.ré cza: is another constRt. Ja the particular case H = 0 the bmier Lsthe same i.n dtlter Oedion, aad these two constants must be the same.

94 ANISOTROPY AND TTMA EFFECTS

Ja this c- it is more convenient to consider the relazation time .r, whichis the average time it takes the system to jump from one miimtlm to theother, instead of the probability of this jump per unit ume. One is tbereiprocal of the other, and the predous equations may be rewdtten (forH = 0) as

1 -. A%V'- = Ae M:,h a = , (5.2.21)r *T

where S îs a constant that has the dîmensions of frequenc'p The originaletimate of Ntxel was h = 109 s-' but recently it hxs become more cus-tomary tzl take jz = 10$0 s-1. Of course, this constant is not necessarilythe same for dsferent ferromagnetic materials.

Strictly spenn'og, cw and c2l (or A) are Onstants only if the mMne-.tization em.naot ever be at azty other angle 0 aad is alwars in one of thetwo enea'v TnA'n'xm a. 1$ rxn only happen if the zninima have zero widths.ïn any realistic caze, there is a fnite probablty of spending some cf thetime in the virqnl' ty of eithez rninimlxvn , izt whicb case the pre-exponentialœelcients ctz and caz are functions of the temperature and of the appliedfeld H. Eowever, if the minima are rather narrow aad the barrier enerais rather largej it em'n be exwcted that ezz aud ozz (or .Jc) have only a'lnenk dependence on T az!d H, whic,h is negëgiblewhen compared with thedepMdence ia the exponentialv and only a small error is ttroduced whenthey are taken as constants. More generally, 'the Kxme On (5.2Q1) shouldalso apply to othe.r ldnds of anisotropy, when .Kzy is replaeod by the erzerppMrrfer for that partimtlar r'xql. The derivation of this muation msK== ed apartkular .form for the bxrrlez %v - &, and it is obvious tllat it does notapply cs it is to other barriers. Strangely enougN, this trivial sta*ment hndto be emphxqixed (:70) because some workers used eqn (5.2.21) for oth.erxn-tscxtropiu. But if the (mrect enersy barrier is used, eqn (5.2.21) holds,prodded that the minima are ratàer narrow and the barrier is zather high.

Tids argument about narrow msnsmx was made more quanuutive byBrowa (171j, who considered the magnetization vector ia a pardcle to wig-le around an enerv minimum for a wve, thea' jnmp (lom wherever it

happens to be then) to somewhere around the other minimum, then wigglearound there before jumping again.. it is Rtuatly a nmdor?z 'tze problemand Brown wrote a d'lFerential equadon to descdbe it, and showed that theeigeavalues of that equation should de*rmine more rigorously the above-mentioned >la and >zl , or r.

Browa did not solve his dœerential equation. lnstead he (171j triedsomeanalyticapprovimations and a.n uymptotic expansion, which he (172)improved later. R'om these estimato he concluded that for a 'lnlid='nlanisotropy the exact solution would not l)e drastiezuly dxerent 1om whatis obYned by tnlring cw and eaa as const=ts) in the range of vatues of thephysical parameters for wlzic,h tlds theozy is usually applied. Numerical so-

SIHRPAMMAGNETISM 95

lutions of Bràwa's diferenti/ equation for the cmse of uninvixl aaisotropy,in zero (173) oz non-zer: (174) appied feldz showed that mssuminr c12 aadcaz to be one aad the same constant is a suldently good appromntion,foz all practical purposes. However, it do% not complicata any nnnlysis ofdaV if a lzighe.r accuracy is used, for whic,h case it is better to adopt atlemst the asmptcdc raatt of Brown, and instead of just a constant h take

2./% a.fa = - forM, rr (5.2.22)

where 'yo is thoe gyromagnetic ratio. For thecase when even better nzvru'raztyis requized, there are several easy-to-use approximations (175) for the tevnrtnumerical solution. Studies of othe,r cases were reviewed in (1762.

The situition is Ompletely d'eerext in the caae of a ctlsic anisotropy. Aslight complication is encountered i.a a calculation similar to that leadtg toeqns (5.2.18) and (5.2.20) here, whic,h calls for a solution of a cubic equationto evaluate 9 at the rnacrimum enera. But at least in the particular caseH = 0 the sohtioa is stzaightfozwazd, 1 '

to a result whicà is VeZ'y m'm-

ilar to eqn (5.2.21) for the uninvlxl case, with the only diFerence that Ah isreplaced by A%/4. However, in this caœ the assumptioa of a constant hctorin fzont of the exponential turns out to be a bad apprrvin/atiom There areminima along the z-, y- aad z- axes (for a positive A%) ='11 very manypn-'bilities of wiglng azoud <>An% one of them before jumping to one ofthe othezs. Kddently, this wealth of possibities makes a big dfereace inthe random-wm problem. A numezical solution (177, 178) for cubic ma-terials gave results which were considerably dsfewmt 10n1 the qlmple Née.lGponential of eqn (5.2.21). Moreover, this diFerenœ is m--mt.rable, becausetàe zelaxation fn'rne eztn be estimated Fom the lino-width of the M6ssbauerspectrum . suc,lz m=uzements for diFerent sizœ of xme cubic partidœat difezent %mperatures weze (1791 ver.r far 1om the prediction of mn(5.2.21) aad quite dose to that which ha.s been obteed by the numericalsolution (1771 of the Brown dîFerential equation. For the accuracy usedhere, this diference will be ignored aad eqn (5.2.27) (with Kzjn hstead ofA%) will be tlsed fcr cubic symmetry too, because s'ucN details are beyondthe scope of tMs book. There aze othe,r approfmations anyway, e.g. theAsumption tlzat the Nrtides are sphere with no shape xn'tsotropy is notalways (170) justiîed. Besidesj uader certain drcumstances the aumptionthat the mavetization in the Ntide is nnlform aad does not depend on

spacey may not be (180) jusMed either. Calculations itave aISO been ze-porte (176) for more complicated cases, such as the case of a magneticfeld applied at an angle t.o the easy anisotropy a='q. These fne details arethorougàly described ia (176J and ws.ll be îaored here.

At any rate, the dependence of the relaxation time on the particle :z.eis in the exwaent, aad an exponential dependence is a very strong one. ln

96 AMSOTROPY AND TM UFECTS

Table s-1 Rvamples of thc relxxxtion time r ofspherical particlc, whox radius is R, for two ma-terials at room temperature.

Matedal R (A) r (s) .Cobalt 44 '6 x 105

36 0.1

Iron 140 1.5. x 10S

115 0.07

order to d=onstrate how strong it is, nnmerical exoamplœ are gi'ven for twomateriis, both at room temperature, f.e. with KT = 4.14x 10-14 e Nrg and

6 -zb0th are calculated using eqn (5.2.21) with the N1l value of h = 10 s .

One is huagonal cobalt, for which Kï = 3.9 x 106 erg/cmz. The othezis mtbic izon, whœe eazy axes are along (100), for which Xz = 4.7 x 10Serg/cmS. The vallzes of the rmlnvation time r (in seconds) arelisted in Table5.1., for a cpe-qln Goice of the radlus R of the particle, assumed to be asphere.

The radii in the table are chosen to demonstrate that within a rather'small raage of partide she the relxwtiontime rxn chaagelom beiag mnchlarger to much smaller than a,u arbitrnm-ly chcfe.n time-scale of 100 seonds.A aiFerent'vallze of % would not càange the general form, and would onlyreqlzire slctly dfezent radii to demonsirate the urne poini. A dferentmaaetic material (namely, a diWhwmt viue of A%) would shift the radiivalue at whicà tMs transiuon oezmm, bat it would again show the samefeature of quhe a sharp change fom hrge to smnll vallzes of r when theparticle size ks decreased. It may thus be conclude that the behaviouz offezromagnets depends on the particle size, and may 1>e disectly d-tlereatfor dslerent samplc made of the qxme materlnl. It caa rk!pn be doncludedthat measlzremeuts may sometimes yield dxeerent ruults for the same sam-ple, if they do not take the same tirne. It is thus necessazy to take intoaccolznt the time-scale of the eoeriment, or the experimental time, texp.

If r > fexp, no change of the mavetization can be .observed during thetime of the measurement, and for all pradical pnrposu the magnçœationdoe.s not change with Mme- This Ls the region of stable #erromcretibpkIf a magnetic measurement takes something of the orde,r of secondsl itis seen 9om Table 5.1 that for iron made of particles whœe raius is û1leasi 150â, no change cltn be obseaed duzing the e-xpeHment. ln fact,no chxnge witl be observed in suc.h a sample of iron even if it is keptfor several days. H this size range, almost everytking mentioned in thissection may be ignored. The only point which may not be Wored is that

SUPERPARAMAGNETISM

this stability of the magnetization does not necessarily hold at the Lome.stenergy minimum. lf it is brought by scme means tc the hhh.er m-mimumof Fig. 5.1, it will jlzst s'tay there, practir-qlly for evea', or =:11 it is broughtdown by an appropriate application of a magnetic Eeld. This is the essentialpart of the hystevewis observed in all feromagnets. Jt is important to bearin mind that the etstence of hysteresis means that it is not suëdent tocalculate the lowest eneror of a ferromagnetic system. lt is always possiblethat a lower-energy state eists, but it is not accessible because the systemis stuck in a higher-energy state.

Of course, the scale of l00 s is just a.n Slustration, and for certain ex-periments, oz applications, the scale Tnlty l:e completely diee,rent. '.l'hasfor '--mple, if R is rcquixed that the information on a maaetic tape iskept for veavs, it is neessau to ensure that the particles in the tape arelarge enough to malce 'r > 108 s. J.n studyug (181, 182) rock magnetism,it is necessau to take into account the decay of the ma>etization dur-ing geolo/cal times, which may be Tnx-llx-ons of years. On the other hand,in Mössbauer eect measumments the (experimental time' is the time ofthe Larmor precession, which is of the order of 10-8 s. lt is thus possiblethat partidœ of a œrtain size may be stable for the MKssbauer KeC't butunstable for œnventional maaetic measurements; and samplc whch arestable during a human life-ume may change duriag gKlogical times. Theprinciple is the Kxme, but the time-scale may be diferent, which mlty shiftthe transition, of which only a,n Altmple is give,n in 'Table 5.1.

In the other axtreme, when the partides are small enough to malcer < texp: many fips back and forth of the magnethation occur during thetime of the experiment. Therefore, in zero applied feld the mexqured, awerage value will l:e zero. Ia a non-zezo feld, the thtvmal fuctuadons havetheir way and oore the anisotropy altogether. The calculadon of section5.1 then applies and the average mvnedzation is gien by the Langeviaftmction, ms in eqns (5.1.1)-(5.1.4). The bebaviour is tke Kxme as that ofthe paramaaetic atoms discussed h secdon 2.1, with no hysteresis butwith saturation, whick is reached when all the pazticla ale aligned. Fmztbparticle in this size range behaves like a huge atom, with the spin numberS of the order of 103 or even 104, instead of S of the order of 1 in conven-tional parxmagnets. Since the argument çtî the Brzlonin or the Langevinfundion 1s. proportional to SH, saturation ks revhed in such materials inEelds which are very easy to obtain, whereas in more conventional paromagnets saturation requires vezy high felds, whic,h are often beyond thecapability of the most powerlul magnets available. For this reason, this phe-nomenon of the loss of ferromagnetism in small particles became Hown ass'upeoaumagneilm, when the Ssuper? part was taken to mean tlarge' as i.nsuperonductivity.'

A single partide of such a small size cannot be made' or handled. Ex-peziments are therefore carried out on an ensemble of particles, which in

98

most cmsc have a wide distribution of particle sizu. Such Nrticles wouldgtve rise to a superposition of Lamgevh Gmcdons with dferent valuœof p' = Ms? in the argument, and the meastu.ed fmrve colzld not pxsi-b1y look 111* the Laugevin function. However) since the argument in eqn(5.1.4) contains the fdd .S' as S/T, when the measured maaetization isplotted ms a function of S/T, data for diFerent temperaturœ should su-perimpose cmto one curve. Therefore, the superposition of M.e ns. H1Tonto one curve, and tàe absenœ of hysteresis, used to be taken aa a,a in-dkation that the sample is supeerpaznmn.aetic, evem whe,a khat curve didnot look like a Langevin fundion. With improvu tsdmiquœ for produegvezy small parddœ, their size distribution has become narrow enough for apuze Laagevin fancdon (1%) to be observed, a=d this indirect azgument isnot necessaz'y aay more. The calculation of section 5.1 rAn now be said tq%ve been con6rmed by direct expevlment. Of couDe, a Langevin function(or any other slzmnxr fanction) eltn alnmys be ftted to such data (184, 185)for a rather narrow temperature raage, but the remarubly narrow distri-bution of (183) r>n be f tted to suc.h a function over a mide temwraturerange- h this repect this experiment is still unique iz the literature.

The argument of eqn (5.2.21) xl= convins the temperature in the de-noMnator. The dependence is actually not Just on the pardde six, but on

VjT. Therefoa the transition fzom stable ferromagnetism to sup -

agnetism, which is demonstrated ia Table 5.1 for the c,ase of room tempera-ture) sh-tfks to a smaller pmicle size when the temperature is decreased- Inmeasurements of the MH r:- S/T curve, some hysteresis app/xztm suddenlyat a suEdently low temperatureo when the sample becomes a feromaaet.Nattzrally, the data at thœe 1ow temperatures are excluded î186) *om thesuperpositiom Tke temperature at which suc,h a transidon occurs, namdytkat for which the relaxation time r Ls e>cl to the time of tke expdm'menttvp, ks e-xlled the bWkgn.q ferrlwmtvq Ta . lf there Ls a size distributton inthe sample, the same temNature may sometees l:e above Tz for some ofthe particlu and below Ts for txe others. Sueh a Kxmple may thus lxk snz-

aaedc for some lligh vtkluœ of the temperature T, ferromagneticat low value cf T, azd a mx''rtkta cf :0th at iutermedîate T. A demonstra-tion of this efect eztn be seen ia Fig. 3 of (1871, which plots the Mössbauerefect data for the same sample at difexent taperatuzes. At T = 5K thereis .a pare six-lines structure of a ferromagnet. At T = 324K there is onecenkal line of a paramagnet, amd at the in-betWeen temperatures there isaa obvious mixqng of b0th, with the superparamaaetic portion increasiagwith incremsing temperatmea

' This pattern ks very m'vnslnm to the changes in the Mössbauer spectrumobsewed at the same temperatare when the averagepaztclesizeîs changed.Ydeed, if tke properties depend on T/F, the efed of cAaaging F shouldbe the same as that of nhnm#ng T. An illustration of the efect of vazyingthe ee at a constant T rAn be sœn for evnmple ia Fig. 3 of (188), which

USOTROPY AND TM EEFRCTS

SWERPARAMAGNETISM 99

actually represents a materx that is an anhlenmmagnet tand not a fer-romagnet) for a large particle =-e,p or 1ow temperaturœ. This experimeat(as well as others) shows that the argument used here applies as well toaatiferromxgnets, whicx n.1M beeome superparamagnets when the pvticlesize is Kmail enough for the thermal f uctuations to dip the maaetizationback and forth during the time of the '

ent. TMs G?.C't is quite obvi-ous fzom the derivation in the foregoing, and it ks n.1M dpne that the s=ea'pplies (186) to ferrimagnets, but it ts always nicer to have an azpezimenfalvprif cation for any theretical condusion. The same pattern of a tnansitionfrom one to six lines eAn also be obtained by the application :189) of vadousmagnetic Eelds. lt lzas alrp-'tdy ben mentioned that the f .eld scaze of SHmnlres it possible to reac,h the - eut of all pardcles at easily attainedfelds. Therefore, it is possibleglg) to se the whole developmeat from zzeroto partial to a tot/ aliglzment, and this chxnge wit,h the app:ed feld isquite ssrnslnm to the paûern Gange with cxanging tempeature-

The wide distribu'tion of particle sizes is most probably the rnm'n re%onfor the gradnd disapp-ance of hysteresis when the c'cenue particle sizedee-rex-- The sharp chxnge predicted by the theory is s'mecred in meaeure-meats g190) of the hysteresis propertiœ (ï.e. remaneace and coercivity) of '

Sessentially spheriœ' particles as a function of iheir median diameters. Ofcourse, whea the sample contaias biggeramd smatler pardcliw someof themmay be ferromagnets aad some pnmmagaets at a certain tempbrature, andthe measuze propertiœ will then show some sort of a partial hysteresis,as i.a the M6ssbauer eâ'ect data mGtioned in the foregoing. However, it ispossible that part of tlzis gradual changG at lemst ia the comrtn'vity, mayix due to a dl'Ferent Kect. When a particle is rnxaethed along +z, ittxkns a feld H = Hc to reverse its magneœation (see Fig. 1-1)- If a feldE < Hc is applied, theee is aa enerr barrier, whic,h is clso zzwporlitmcî tofhc 'tlolumet that preven? the reversal, If the particle size îs a little abovethat whic,h allows a spontaneous Sipj it may lp anyway at a Eeld wkick issomewhat below the bulk coercivity.

There are many theorio of suc,h a meclmnirn in uncial partide (191)192, 19% 194) or pla/lets (195) and some attempts to take it into accouat(196) in numerical simuhtions oî 'tàe mn.aettzation process. There is evensome 4x-efm' 'ate (197) for the-tllMnltl fuctuations overcoming a diferentldnd of enera bxrrier, for the motion of a domain wa2 îa bigger particleswhich are subdivided into domains. However, none of these theories hasever been suEciendy developed for even tem-ng l these efcrts are largeor small for any rpABstic case. And none of them !mn ever rearlled theasuge of wondering about the raadom walk of the magnethation whic,h hubeen mentioned oxrlier in lbl-s secdon and whîch has not been properlysolved evea for m-mpler cxqx. It slkould ix particularly emphasized thatmost of the (eeriments mentione in 1%!K section are semi-quatitative,and CIIeCIC only some features of the theorp The thKry predids a 1- of

10o

the ferromaaetksm when the particle is small or the temperature is high,and t/19 plection is ceruînly confzhed. But this theory remninq quitecrude, and IUA not been developed into more accurate estimates, becausethee are no erperiments that call for a higher accuracy.Actually, except forex-periments suck as (179) and some of those discussed ia (1805, there ks verylittle comparison of experiments with guantitadne theoretical predictionsof what the relaxation tlme is and where the kansition should odcur inreal matezials. The maln reuon is that a qllxmtitative exxriment is verydilcult to c'arry out, as *1 be discussu in the next sectîon.

ANISOTROPYAND 'lqA EFFEGTS

5-3 Magnetic ViscosityBetwen the Kie,n of supeparamngnetism and that of stable feromagnetismthere is in prindple a particle size for whic.h r is of the order of texp.According to the ev-trn ple in Table 5.1 it is a very narrow size range, andit is usually quita difcult to prepare a sxrnple of the necusary size to seewhat happens thea. For some teeniques of maldng small particleq thesize distribution may well be larger thaa this transition region. Moreover,it is aot even always possible to measure the size of thee. particles, so

' much so that there have been maay suggetîoms aud attempts to uœ thesuperparamagaetic transition aa a ct-nre for the distributîon (189, 198)or at least the anevage (182, 19$ of the particle size. Suc.h memsurementsobviously call for a bette,r theoretical interpretation tha.n the oversimplledutimate of the predous section, WhicA nxqumes that all the spins withine-qzt!t particle àre aliaed. Besides other challenges (180) to this assumption,the mere fnrt thtat a large propordon of the spins is near the surhce insuch small particles shoald makn one suspldous of aay theor.r that doesnot take 1.a10 account the prm'ble efec't of the surface anisotropy.

In practice there is very strong evidence (187) 189, 200) 201, 202) 2031that the magnedzation near the surhce is oftea quite dsgerent from that inthe A-nner part of the particle (- also the last pasrlgrapk of section 5.1.5).Iron paMicles in particdar may be ofdized, so that they are Ktuazy madeof an ironcore srounded by a shell of iron ofde (201, 204, 2051, for whichthe'simple theory of the previous section does not apply. Surfve Gect,s =n.yalso be implied from the observationtzo6) that magnetic propertie of smallparticles are sometimu sensitive to surfactants adsorbed on the sner.p. Ithaa also been noticed (207) that the shape of Ge pmicles =ny not bespherical) and that they may tead to stick togetherj forming loag chains(208) or other (209) aggxzgates, which change (210, 211, 212) the relaxationtime condderably. InterKtions beœeen particlO have been demoastrated(213, 214) to be veayimportant in real ieasurements, aad these interactionsno.y sometîmes look lilce a 8ze dkstdbution (215) in aaalysing MössbauereFCSC't data. Other efects, sucA as magnetostriction, may also be interpretedas if they were (215) a size distribution. There is thus little wonde thatthe particle size determined fzom the magnetic measurements rAn oftea

MAGNETIC VISCOSITY l01

be very dllerent (216) 1om their directly meazured sizer although bothsucâ meuurements av6 sometime (2171 consistent, for kuite uaiform andwemisolated' pariidœ.

Comparison betwen theory and experiment in this particular feld isfurther complicated by the unuown physical constants, because b0th thesatuzation magnetization (217, 218, 219) and the anisotropy constant (217,220) of :ne particle dfer fzom thdr bulk values. lf these parameters are

adjusted for the smatl particles, thee is not really any direct evaluation ofwhat the theory of the preuous sedion predicts. The Curie temperaturernlky n.1M be d-lfFerent for smatl partides (221, 222) fzom what it is in thebulk, or there may be some small resons within the particle (223) which Epbefore the magnethation of the whole parhcle fips, when the Curie pointis approacked. And atl these unHowns aad uncertaindes are supmo-mpceedon a theory whic.h is exiremely sensitive to =ull mistake in the partidesize (224), or ia other physical parameters, ms demonstrated in Table 5.1.

ln spite of a11 these dMculties, there is a surpridngly large number ofexperiments in the literature for partides in this narrow reon for wizickr ;4$ toxp, even though it is not clpxr i.a mxny c,ascs whether it is rallythe whole sample, or only Nrt of it, for whicà the particle are in thissize range. In this re#on of 'r, the menetic pxoperties change while beingme%ured, and this ckaage can in principle be observed. Thus, for Altmple,if a magnetïc feld is applied and then r-oved, the average menehzationdecays on a time-scale of the order of T, which should be possible to mea-sure- A decay is usually exponential to a frst order, so tlat the r=anentmagnetizadon should luy-bn.ve according to

.M' (.j) = .<r(O) e-t/.r (5.3.23)r ,

where t is the time- Plttiztg experimental data to this relation eltn yield thevalue of the Telnavxtion time, r, or at leut its average when the system hasa distribudon of the valuœ çtî r.

. Eowever,nobody evertries to ft data to eqn (5.3.23), because it is takenforgrantedthat there must be a wide distribution of tkepartïcie sizel whichmust camse a wide distribution of r, and tke time decay is actually

-

,(r)e-'/- dr,Afr @) = Mr(0) jo (5.3.24)

where P is a distribution function. O1d estimations, and more recent nu-merical computations (225) for snecisc distribution functîons P, show thatunder certain coaditions eqn (5.3.M) e-xn be appzoimated by

Mr(à) = & - Sh(t/a), (5.3.25)where C and S aTe constants, and this functional form ks used to analyse

102 AFRSOTROPY AND TIME EEFECTS

>

logtlz-olFIG. 5.2. Schematic representation of a magnetization decay on a loga-

rithmic scale.

practirluly all experimental data- Most workers omit the conslmnt a andabsorb it in Ct but it is wrong to do so, bec-ause a logadthm is only defnedfoz a dimensionless number. It may also be worth noting that most workezsdo not choose the integrand as in eqn (5.3.24), with a distribution of thevalues of r. They prefer a distribution of the particle sizes or of the energybarriers, aad use the dubious assumption that the relation between theseparameters and 'r is Hown and established. It is not) according to thediscussion in the previous section.

This choice of the logaatthmic function is rather strange, because itis not regular for either small or large values of t, and can certainly notrepresent the be ' '

g or the end of the memsurements.' The real functionmay at most be linear in lult/al over a ll-mfted range, which does notcontoân the short and the long time. ln pzinciple, it can at most look likethe schematic plot in Fig. 5.2, and indeed tMs form is what is observed(226) when data aa-e tAken over a wide range of the time. However, m=ye'xperimentalists just assume that the logarithm is the Ctrue' form to beused, and they do not report (or do not measure) aqything outside therange for which eqn (5.3.25) can be ftted. Cases have been reviewed (227)in Whic,k the reported time range was so narrow that it may not even be inthe linear re#on of Fig. 5.2, aad in an extreme case Mr was measured atonly t'tvo 'tzcl'ue..s of t, hz ordez to determine S of eqn (5.3.25).

The logarithm is so inconvenient that even if it were an essentia) part ofthe physical problem, there should be some attempts to avoid it as much aspossible. Using it as an approfmation, e'an ifit is a good approfmation, asis cllurned (225) for certaîn c-ases, is a completely unnecessary complication.It has been claimed (228) that the criticism of the logarithmic function as

MAGNETIC WSCOSITY lO3

brfu.king down for large aztd smallt is Vcorrect' (.Wc!), becattse eqn (5.3.25)is only used for a certain time-window tzzun << t << tmax: but no reason wasever given for leaving out the time outside tids regiom Actually, the limits

' of the re#on which is linear i.:z ln(@/erb) are related (229) to the 'width of thedistribution. By trying to ft everytlting to eqn (5.3.25): or by avoiding there#on outside that time-window, important physical information is thuslost. Moreover, there is reason to believe that in many experiments onlyycrf of the sample decays with r of the order of àexp. The main justlcationfor using eqn (5.3.25) ks (225J that it is a good approfmation ffor a widedistribution of eaergy barriers'. However) if the distribation is wide, it iseasy to imagine that some of the particles are large enough to be stableferromagnets under the condîtions of the expeHment or that some of theparticles are small enough to be superparamagnetic: or b0th. teaving outthe part of the decay curve for short and for long values of the time leavuout all the information about the small and the large particles in the >m-plc. It is a risky procedure, especiany since it has Yen shown that at leastone method produces (230) two groups of pazj icles in the same sample: largeones which are ferromagnetic and small ones wlkich are superparamagnetic.On top of al1 thatj a logarithm may not even be a true representation ofa wide distribution, bec-ause an cltezwtzfse explanation (229) says that a.napparent linear dependence on log//al may also be caused by magneto-static interactions among the particles. As long as eqn (5.3.25) 'is used forthe analysis of the data, it b impossible to distinguish between these twoefects. .

Actually, it is not even necessary to look for an approimation which iseasier to use than the logarithm) because it is possible (227) to carry outthe integration in eqn (5.3.24) rigorously and ualyticallyj if P(m) Ls takento be the so-called gamma distribution Jurlctforl,

(5.3.2 6)

where r is the gamma ftmctiozb and ;) and eo are adjustable parameters.This function looks more or less like any other probability ftmctiozb as canbe seen from the three examples plotted in Fig. 5.3 for the particular choiceof ;) = 2, 3 and 4. On this reduced scale, the value of a does not have tobe spedfed, but ït will obviously have to be îf r ïs given in real unif.s ofdme. Graphs exn readily be plotted for other values of these parameters,and thiy all look qualitatively the same. lt 1s, therefore: ms le#timate touse as any other distribution ftmction) and at lemst no convindng argumenthms ever been presented for the use of any other distribution function: thechoice of which is also quite arbitrary. My diserence between diferentprobabiity functions ks at most a second-order efectj which is better leftto be studied only after a1.1 the frst-order esects have been clnrifled. The

1 'r P-Z

P(S = azxtp; (-a e-M'IM

,

104 AMSOTROPY AND rrf> KFFECTSf-C

n%

bqo

/ Nv

l ,-N -.

t'q /? .. .-& e . x .s

;e q? , x -.

, '/ . N *.

k e y ë 'h' !;

r Nu -./ . -.

X ' 2 Y% -q.

ç;' 4 / - -xv -.* <

I ? .. --

.h-j . h.: / .

I / .e k& '

($ c 4 5s/roFltI. 5.3. The gxmma distribution funcdon of eqn (5.3.26) plotted for

;) = 2 (6111 curve), p = 3 (dashed curve) and p = 4 (dotted curve).

same function of eqn (5.3.24) was ugM (231) for a distribution of enervbarriers, iastead of tke distribution of relavxdon tima used here. For thisrx-ee the intqgration cazmot be nltrrled out analytically as is done here.

When eqn (5.3.26) ks substitute in eqn (5.3.24) and the integratioa iscarried out, the result ks

plkMr ( t ) . 2 ( ..t j xp ( a y .)Mr(0) P@) rc c

(5.3-27)

where Kp ks the modïed Bèssel function of the third lrin d. TV f'unctlonLs well-desned, its properties have been invatigated for any range of theparxmeters, and zPKp(z4 has no sinalarities. Therefore, all sorts of e.x-perimental data may be stted to this function, aad no separate treatmentis needed for large or small @. Suc,h a Etting should determine t'he two pa-rameters, p and a, of the fanction P of eqn (5.3.26). Their vallzes thendetermiae the two pazameters which are most signifnxmt for any physiciproblem t'hat izwolves aay ldnd of probabilities. One of them is the mannalne, whicb ia the case of eqn (5.3.26) is given by

m = lrb , (5.3.28)and the other Ls the rcrïcxez, which in this case is

0.2 = mj . (5.3.29)

TM STONER-WOMFARZH MODBL 105

This important physioal infozmation about the pazticular system understudy is just lœt if eqn (5.3.25) is used.

The pœsibiiit of using the gamma distribution function hms somehowbeen ignored, and ig not even mentioned in the most recent review (225) ofdferent models. They (2251 and others (232) insist on choosing some otherP('r), and - out the integratioa numerically. And even for cases inwhicx logtt/a) turns out to be inadequate, they (225) and others (228) sug-gest using a Jmcer series in loglval, thus conserving the inconvenient andnon-physical siagalarlty for small aad large @. Similar suggestions have beenreviewed in (227). There ha.s been an attempt (233) to plot one universalcuzve for the decay of the magnetization measured for the same sample atdiferent temperaturœ T. However, even for that purpose, it was suggestedthat the data be plotted ms a bnction of (T/Tc) 1:1(Va), wMle R is clearthat any fundion of that parameter ks also a Iunction of teTlvo - lt ceemsthat this 5e1d crnot advance befoze the obsession with loguthms is over.

5.4 The Stoner-Wohlfarth ModelWhe,n a fœromagneticpazticleis lazgeenough, all the kime-eeds dœcribedin section 5.2 are neglisbly small- Nevertheless: such particles may still besmall enough for the exchaage enerr to hold a11 spins tightly paralld toeach other, and not allow the space-dependence of the maaetization whichenters only at a larger particle size. ln tlds =e, a.s in the case studied insedion 5.% tAe excxange enerr is a constant, and de not enter theenergy mlnimizations. There are then only the anisotropy energy of theparticle and the interaction with the appBed Eeld to be considered. It isthen mssible to use the same enerr relation a.s in eqn (5.2.13) to solve forthe hystaresis curve of these svble but smatl ferromagnetic particlœ. Suciza calczzlation is knx ms the Stoner-Wohlfavth modet

Actually, the original study (2311 of Stoner and Wohlfxrih assumed ashape anisotropswhick .111 be dMned in section 6.1, and not the unip.vlxl,crystuine xniuropy as in eqn (5Q.13). Howeverj the mathematics is thesame, aad thks model was also used later for the cxse of this anisotropy.Moreover, acalculation based on this model k beiag widely used à) measurethe erpsïiline anisotropy, as mendoned in sectlon 5.1.6.

'ne mm-n mssumpdon of Stonc aad Woidarth is that the materialis made up of rather small particles, which az'e suëdeùtly separated fromelmlk other so that interactions between them are negYble- Ifthe magneticEeld, H, is applied at an angle I to the easy azs of the nnia'dal xnlsotropyof the particle, the magnetization vector will rotate to a.n aagle $ fromthe Eeld diredion, which means that the maaetization vill be at an angle4 - e from the easy aHs. The energy of this system is the same ms ix eqn(5.2.13)7 with the change of the anglœ into the onœ deîned here, nnmely

17 = A'zk' sin2(4 - 0) - /zJ./ cosy, (5.4.30)

l06 ANISOTROPY Ar TTMR BEFECTS

where F is the volume, aad whee the maaetic momeat (b of the partidemay lx replaced by MsV, as was later done in seGon 5.2 as well.

Stoner and Wohlfaztk preferre to use a diferent dïdniton of the e.ne+zero, aad replRed sx by the cosine of the double ngle. They worked withthe reduced energy,

'

s cos(2(4 - 8))n = + const = - - à, cos/jzAtF 4 (5.4.31)

whceMsFh = . (5.4.32)2A%

For gkven valuu of 0 aad àotbe maRelization will choosethe augle $ whic.bmînstnl'zes this enœrl n=dy the solution of

...3.8 1= g sin(2(4 - #)) + ?zsin $ = 0,t9/ (5-4-33)

provided that the solution represeats an energy minîmum and not a maf-m=. This conditicn c.aa be e-xpressed 'as

= cos(2(4 - 8)) + Acos/ > o.:42 (5.4.34)

Be ause of the muld-valued trigonome'c functions, eqn (5.4.33) hasalways more thaa one solution for a tven ?& aud 9, and it can happen thatmore than one of tlve xlutions represeni an energy minimum. H order toobtain a unique solution., it is necessazyto specify, aad follow, the history oftàe value of h for e,ae,h 8. A Klution wlzic,h starts 1om a particular branchc=not be just allowe to jump into anoier brach. The Jump must be ata feld value at whiG there is no enerr bnrrier between these brancke.Tkks importan.t feature is the basis of the hysteruis which is always partof mMnetism, aad in order to see how it works it helps to look flrst at thetrivial case 8 = 0. In this case eqns (5.4.33) and (5.4.34) a2e

(h + cos /) sin / = 0 and cos(24) + Acos/ > 0. (5.4.35)One soladon of the ârst half is cos $ = -h, whiex Ls a valid soludon if1:1 < 1, but it do% not A4161 the second hnlf . '7%l's xlution rep=ents anenergy mcdrn'um and has no physiœl sigGczmce. The other solution Ls

sin 4 = % axld 1 + ?zcos 4 > 0. (5.4.36)The combhation mpltnq that it is necessary to use $ = 0 for h > -1, aad$ = zr for à, < 1.

TIœ STONER-WOIKEAETH MODEL 107

It is thus s-n that the Klution is unique if 1?zI > 1, but in the re#on1à1 < 1 both 4 = 0 and 4 = 'r az'e valid energy minima. At this point itis necessary to introduce the âe2d Mstory. If we siart by applying a largepositive ltt then reduce the feld to zero, and hcrease it in the opposite di-rection, the physical system remains on the branch of the solution 4 = 0 tillthe feld h = -1 is reached. At this âeld the solution bccomes unstable, andthe system must jump to the other brancb, 4 = x. Note ia particular thataccording to eqn (5.4.31) the reduced enera in tMs cJue is n = -1-h,cos4-4Once h pa%ses zero, and becomes even slightly negadve; the state 4 = 0 hapsa higher enerr than that with $ = x. However, the maretization emmnotjust jump into the lower-enera state, because it is in a rnin-lmtlm eneraAute, which mp-xns that there is an energy barrier tllat holds it there. TEesituauon is m'z'nilar to the enera displayed in Fig. 5.1. The system is juststuck in the b-lrher-enerr state till the feld reaches the value h = -I, atwhic,h the bnarryr is removed and a Jump to a lower-enera state becomespossible. A similar, but reversed, argumeat appliœ to starting Fom a largenegative A,A in which r-qme the other brancb is held till the feld reaches tkevalue lt = 1. The whole hysteresis curve is then qnalitatively m'moar to thel'lmiting cmwe plotted ia ng. 1.1, aud the coercivity ms defned there isfor the reduced feld h = 1, which mexns Hc = 2KïfMs according to mn(5.4.32) .

'

If 9 # 0, eqn (5.4.33) has to be solved nnrneriexllyl but the' general be-haviour is rather nimilar to tke cxqe of # = 0 which has just been described.Starting 1om a lvge positive Geld, the solution which starts with $ = 0,ï.e- cos/ = 1, curves down with deceasing values of h, to lower values ofc09, nn.mely to smaller valuo of tlle component of the maaetization inthe âeld .dizectiony

Mu = Ms =ss. (5.4.37)At tke poiat wkere this branch stops to be a minimum, there is a jumpto a second branch, thus displaying sometMng which looks more or lesslike Fig. 1.1. Obviously, the jump occurs where the left h=d dde of eqn(5.4.34) paasc through zezo, maldng that branch changefrom a msnimamto a memum. The combination of a zero for tMs muadon together witheqn (5-4.33) givc rise to several reladons g2341 between the kritical' valuesof h' a'ad 4 at which the jurnp occurs for a #vea 9.

It may be interesting to bok azso at the other extreme case whi/does not r.n.ll for a numerical eeuation. Tkis case is 9 = x/2, ï,e, a îeldperpendicular to the exsy afs of the aïsotropy, which efectively meaMno anisotropy at a11. In this cœse eqns (5.4.33) and (5.4.34) become

Lh - cos 4) sin 4 = O and - cos(24) + h' cos/ > 0. (5.4.38)In tMs case, the solution cos 4 = h, whicb is a valid soluuon if 1à1 < 1,also fulfls the second half of eqn (5.4.38), and is an e.ne.. minimum. It

108

yields a magaeœadon pzoportional to the feld, as ia a pazamagnet, withno hysterois ald with zero coercivity. At h = +1 it changes ovez to thevond soludon of sin / = 0, wikicll is the saturation of 4 = 0 or 4 = A'.

ATA computhg the hysteresis curves for each feld angle 9, Stoner aadWolllfnvth (234) computed the a'verage for a random distribution of theanglu 0t namely a collection of pardcles with a random distribution of thedirection of their easy aN:% with respect to the direction of the applied îeld.The resulting c'urve is very m-rnilar to the one shown 5n Fig. 1.4. Actually,maay expehmental ctlrv'es could b'e aualysed in terms of this simple theory,which Axq been widely used over the yeats. Even magnetization curvuof thin permalloy fl=n obey approvirnately the Stoner-Wohlfarth theory,although the physical mechanism behnd it is not clear.

The main adwrtage of thks theory Ls that it is suEcieatly m'mple toadd some e-xtza featuru to it. li is jmst as easy to rephce the randomdistzibution of 9 by some other distribution, fvmtre wherever theze is aaexperimental reason to belîeve that the dhetions oî eas'y xws are morelikely to be, a: in the caae of au atigne; or a partly aligaed, maoetictape. The c%e of a cubk, ïn:tetzd of a no-lx=u.l, aaisotropy has alx beenworked out (235) in detail. In this case there are more branGes ixaaa in thetmlaMal case, which makcu,s it sometima more diEcult to deide into whichbranch to jump. But. these diEculties caa be handled. A r=dom cubicanisokopy besides aa overall uniafal one has also been IISM (236) ia thestudy of the magnetizatloniripplel mentioned ia secdon 5.1.4. The parallela'nd perpendlcular susceptibllities (161)j discussed in section 5.1.6, have alsobeen calculatedffrom this model. Fkrther developments and a study of thefner details (23% caa even stvt to oFer a physical intezpretation for theirersv;ce between crp/m-mental rœults aad the Stoner-Wohlfvth theory.It may sometimes leM to an understanding of the parts neglected in theStoner-Wohlfnvkh thers whic,h are the intervtions between the particlesand the possibility of some s ependence of the magnetization wiiin I

each partide. Hteractions of certain goups of ellipsoids have also beencomputed (2381 for this model.

ANISOTROPY Ar 'lqME EFFECTS

6

ANOTHER ENERGY TERM

6.1 Basic MagnetostaticsBesides the enera terms discussed so far, there is Mother term whicb hasnot been meationed yet, acd it iqi Mme to introduce it. This term is tkemannetostaxc setf-encrgp which ori#nates 1om the classical interactionsamong the dipolo. For a continnotta material it is desczibed by Mamell'sequations, whicll the rpmzer is mssumed fo be flns&-llar wità, in the formtaught N) undergraduates, even i.f not necessarily Gmiliar with the partwhich is m-t zelevant for ferromarets. From a historical point of view itis interedng to note that this enerr term was part of the Hrlltoniau inthe early study çtî (39) spin waves, wMch included the anisotropy a: well.Dyson (401 oblected to some oî the approx-imatîons used (39) for tMs term,but did not htrMuce anz othe, and since then somehow eveubody Justgot nsed to leaving out this eaerr term.

Ixt the meaztime it is just ammed for simplicity that the ma%rial iscontinuous, leaving for the ne-x't chxpter the study of a crystal made outof discrete atoms (oz ions). Not atl of ll's equations are use inthe present discussion of a Iemmagnek wi't,h no particular zeferenœ to itselectric properties. One of the equations state that

V x H = 0, (6.1.1)in tàe absence of any currents, or displacement currents. It should l)e noted,hqwever, that this assumption of zero currents doœ not lead to a rœtrictive,paedcular case. Gt is customary in the stady çtî ferzomxgnetksm to separatethe magnetic felds into two categories azd treat the âeld H in eqn (6.1.1)as separate from the appïied Jeld producH by currents in coiks- As longas these tdiferent' felds are properly superimpcee, thee is no loss ofgenerality, and Gere is nothing wrong with tEis ctmtrenfcnl notation.

The most general solution of eqn (6.1.1) is well known. The vector H isa vadient of a scalar, % called the potential. Tàe convention is to defneit with a minus sign,

H = - %U.

Another of Maxwell's equations is

V . B = 0) (6.1.3)

11û

where B ks tàe mannetic fnducfïoG defned ia eqn (1.1.2). Se tb.e book byBrown g1) for the derivaïon of these equations aad for a z'igorous deinitionof the vectors B and E. It should only l)e emphasized that it is nmong lowrite V - E = 0, as is done in some books. The latte.r is equivalent t,o eqn(6.1.3) only if eqn (1.1.3) holdw which is not the case in ferromagaetism.The faztor % invented by Brown (11 will be used throughout this chapter,as a way of iatrodaction, in order to mxlr, the trnnm'tioa pnmie.r for readerswho iave only used the S1 anîts till now. H the SI units, now used in 21undergraduate tutbooks, its value Ls % = 1, while in the Gaassiaa, cgsunits, used in all the literature on magnetism, % = 47r. More conversionfactors are listed in section 6.4, and 1om there on, for the rest of the bookonly she cgs system of units wûl be :RKM . mltly or wroagls tMs systemof units is still lx%tl almost exclusivezy in all the Bterature on rnxretisp,eve.n though some unge of SI is strting to creep into some of the morereent papers. For =ybody who wants to study tMs subject there is noalternative to getting used to tEe cgs units.

Substituting eqns (1.1-2) rd (6.1.2) izt eqn (6-1.3), we obl-qin

V2Uin = CSV . M, (6.1.4)

AIVOTHER ENBRGY TEEM

which should be valid inside the ferromaoetic body (or bees). Outsidethis bod.y (or these bodies) M = 0, so that B = E =d the dz'Ferentialequation is

Vzuout = 0. (6-1-5)It is also Hown 1om undervaduate textbooY that Maxwell's equationsrmuire *at the componeat of H Nmtllel to the surfaceo and the compmeatof B pezpendiculr to the surface, are continuous on the boundary of twomatezials. These requiremeats lead to the boutlazy condtioas that on thesurface of the ferzomMnet,

OUw XFout '

Uin = Wut , o -

sz = 'ysM - zl , (6-1.6)

where n ks the unit normal to the surfu of the ferromagnedc body (orbofes), txknm to be pYtivein the outward direcdoa. Beside thae bound-al'y conditionsj the potential U is required to be regnlav at inMity, wàichmeaas that both Irul and Ir2VtJl are bounded as v ...+ = This regukxrityœsentially meaas that the beàaviour of the potentW at a large distaace*om the magaetized bodies is the same as that of the potential of a pointcltarge, wlticlt rxn be expected if the maaetization vanisho outside a cer-tain fnite volume. . ,

Hstead of the scalar potentiazy the problem may be formulated muallywell by writing B = V x A aud deriving a diferendal equation with bound-ary coaditions for tàe eector Joterztfttl A. However, this formulation is 1-

BASIC MAGNETOSTATICS 111

convenient for the problems dismzssed in tMs book, and will not be usedhme

'

Once the d'-Ferential equations and boundary condisions have beenrelve aad U is known for the whole space, H eAn be calculated fxommn (6.1.2). The ener-?p eltn thea be evaluated as

1su = -y M . H dr, (6-1.7)

where the iategration is ove,r the ferromaaetic bodies. ntq equadon willbe proved more rigorously in the next chapter. 1n the m'Gnume it mny betaken a.a the hteradion of eMh dii o1e with the field H created by the othe.rdipoles, witph a factor à being iatroduced in order to avoid countkg mice?the interaction of .â mth B ) aad pf B wit,h .A-

6.1.1 UnjqneneaThe most important feature of these diferential muadons aud boundarycoadidons is'that their solution is Sznësze. In order to prove this stat-ent,suppose that there are two Snctions of spaœ, Uï and U1, that fulGl allthe equations (6.1.4) to (6.1.6) aad are IIOG r at l-nfnity. Then thefunctioa Uz = Uz - Uc aad its derivative must be continnous evewhere,iacluding the sueces on which the normal derivatives of Ift aad Va arediscontinuoc. Also, acemding to eqn (6.1.4), T2Uz = 0 evezywhere, wbic,hmeaas that for an interalion ove,r aay arbitrary volume,

drzrsTU h 2 dr = 7V . ( (Ja Vé7j A -. rzra /2 (Jzl gr = (Ja d.q( zp L i : on I

where the second muality is a maaifestation of the divergence theorem,aad the 1% iateral ks over the surface surrounding the chosen, arbitraz'y

lmme.' It should be note that such a use of the divergence theorem is notvoallowe for Uk or Ug, because of the discontinuity expressed by eqn (6.1.6),whic,h reqm-rtas iategzation ove,r both faces of each discontinuity suv?n ce.Eowever, according to the present asumptioa, both Uz aad its normalderivative are coniinuous everywhere, aad the iategratioas over b0th fMescance) each osher because of the opposite diredions of n.

:If the volume chosen for the iutegrati6n ia eqn (6.1.8) is now allowed totead to infnity, ds increases as ;P while the regularity condition require

W d Uz to decre'xse at lemst as r-l, soouslon to decrease at least as r- , anthat the surMe integral tends to zero. Eence, the intagral of (VG)2 o-the whole space vazishes. And since the integrand is a squarej which elmnotbe negative anywhere, TUz must vanish eveawheze,. whic,h means tkatUz =const. But a non-zero constant is not reoar at l'nGnlty. Therefore,Ua = 0 everywhere, and UJ x Ua.

There is thus only one possible Klution to the potentik problem of any

ANOTHER BNERGY TERM

geometry aad any distribution of the magnetization. Therefore, it is never

nee- to #ve t;e intermedia.te Meps; or to jusffy in aay other way asolution 1.o a potential problem. If a certain f'unction is guused, or arrivedat by any other mexnK, it Lq suëcient to show that if fuïls the dlFererltlnlœmations and the bonndary czmctitons, because if it is a solution of theproblem, it is always tlte solutlon of that moblem-.lt should be noted,however, that wbsle a magnethation distribution determiae a unique Eeldouîide the fecomaaet, the reverse is not tnze. A measurement of the fe-ldoutside a ferromaaetic body is no1 suldeat (239j to determin.e a unîquemagnetization distribution that creates this feld.

6.1.2 FHdcl EzamplesThe theorem about the uniquemess of the solution allows qlzotiag withoutproof the potential for some simple caae The proof is h substitutingeach of these functions in eqns (6.1.4) to (6.1.6)y and checkng that it % asolution- '

The Mt case ks asphere, wh-e radius is .E, uniformly mMnetized alongthe z-diredion. 1.n this caae, V . M = 0, and ia polar cxrdinates r, #, and4 the diFerential equation become

- 1 ,9 ,') 1 ,? ta 1 :2

(WW/ZT'Z'V -f- r2 s'--u'xlp'lt/sin.sv -V .ra :5..ry # :47 U = 0' (6'1.9)

b0th inside and outside the sphere. Akso, in this cn>

01&n = (V& and M . xt = M. = Ms cos 8, (6.1.10)because Ms is the aagnitude of M- It can be veriîed by substitution that

'r if v S RMsU = % zose x (6.1.11)3 a /ra jf z k aRIMtilk'fles eqa (6-1-9)y is czmtinuous at v = .E, haa the appropriate discontinu-kty of the derivative reqeed by substftutiag eqn (6.1.11) in eqn (6.1.6), aadis regular at inoity. Therefore, it is the solution of the potential probleml'ndde and outside a um*lformly magnethed sphere.

ln particular, the potential inside the sphere is actually

MsUin = % z. (6-1.12)3

Substitutlg iu eqn (6.1.2), the feld inside the sphere is

Hzzn = Hv. = 0 ,Ms

Hzîv = -- % .3 (6.1.13)

BASIC MAGNETOSTATICS 113

lt is, thus, a nnqorrrt feld, wbich is antiparallel to z. Eowever, the z-direction %nA no partimzlar meaning for a sphere, And in the present conterit only denes the irection of the magnetization. Thereforep tlte internalEeld,in a homogeneously magaetized sphere Ls antiparallel to the magne-thation. It should be clear now why spheriocl particles were spedfed iathe previous cxapter. For any other geometue, the direction ofthis intcraalGeld may not be parazel to the easy anisotropy aMs, which complicates theproblem studied there.

The maaetœtatic enem of this uniformly magnethed sphere is ob-tzuned by snbstituting this H in eqn (6.1.7). Since the integrand is a con-staat, the intevation is only a multplicntion by the volume of the spheze,S.g'A3. Therefoa the magnetostatic self-enerr çtî a uiformly magnethedssphere is

27 aEu = -x.!8Ms . (6-1.14)9The second exaaple is an infnite fn'rcttlar c'ylinder which is uniformly

magnetized along the z-ats, where the c'ylinde.r axis is dvned ms z. Arln,V . M = 0 evermhere, and in the cylindrical Oordsnatœ p, 4, and z thediserentlnxl equation is

g 1 (7 (7 1 02 03- p + --5.

-

2 + x D- = 0,I p V Tp J .S / ta z (6.1.15)

while the normal is parallel to p, and the normnl component is

M - n = M: cos $. (6.1.16)It rltn be 'vr>6- ed by subetution that the solution for this case is

p i.f p % .RMsU = nN ct)s $ x (6.1.17)2 a j kf z p.E , p p

where this time R is the rxzh-us of the cylinder. The intlrn al îeld inside thecylinder is

MsH=tv = -

a % , Hujx = Nx'a = () , (6.1.18)whic,h is also a uniform îeld, antiparallel to the magnethation vector. Theenera p6r 'uzzïf length cîtmg z is

'm z ztx = -% R Ms - (6.1.19)4

If the same cylînder is znagnetized along the zdirection, eqn (6.1.15) isstûl the diferential equation to be solved, but in the boundary condidon

114

of.mn (6.1.6) one Gould talrfR M.n = 0. The solution is then U = 0, whichleads to H = 0 and Eu = 0..

6.1.3 Uïujrmly M'tzrletile,d Bllipsoid '

The Mvxmples of a sphere and a cylinder are particalar cmses of a mozegeneral theorem about uniformly magneuzed elpsoids, whicll wms alreadyknown to Maxwell- It will be stated here without proof.

Generallyj the feld l'nm'dc a unlformly magtetized ferromaaetic bodyLs nof u/form. Eowever, if and only if the surface of iàis body Ls of asecond degree, the interaal Eeld ks uniform. This theorem is often statedms applying only t,o ellipsoidsj rather than to snrfxces of a second degreeobecause atl other Mond-degree surfaces exte'nd to infnlty and Amot bereal'tvod in pzactice. Stilk the cpsoid is usually understood to indude the

. *

limitiag cxse of an tnf nste cylinder.When the Cartean coordl'nates are cllœen along the yrfncilcl t?.'rcd of

a general ellipsoid, the equation of its snrfnre is

12 :2 z2

(z) + (,) + (-c) - 1 wîth u s b s c. (6-1.20)

ANOTMR ENKERGY TEEM

lt may sometimes be necessao- to de6ne z, y, and z ia other directionsjbut the rtozier is suppivW io know how to perform the rotation of the :2:6.3in this owation, and in the' ones that follow- If this ellipsoid is unifcrzalymagnetized, the feld inside the ellipsoid rltn be wzitten as

H$a = -N - M = -.'>D ' M, (6.1.21)where both D and N = >D are tevors. In the partictllar cxo whenM is parallel to one of the prindpal axes of the Gipsoid, both D and Nare ntlmzera, and both aze Hown by the name demagnetiping jatocs, orsometime demamt#zatkon Actors. Dcept for the use of the letters D andN, which is almost (btzt not quite) univttmxl, it is sometimo dilcult $otell mltiolt of thoe dema>etizing hctors is bejng referred to. lt should alsobe noted that this feld (A1r.n Hown as the demagnetiziag feld) is tke partceated by the magnetizatton. lf there is also a,n appliqd feld, produMbz some currents in mernal Oils, tkese felds sqxm'mpose and have to l)esumrned vectodxlly. It should rklsxa be noted that all this treatment appliesonly to the case of an dlipsoid which is uniformly magaetized, and not toany other spatial distzibution of the magnetization.

Thelmst partof tMs theorem Ls that the trace of the tensor D is 1, whichalso mpmns that the trz- of N is %. Therefore, the results of the foregoingexamples for the sphere and the cyMder eAn be obtined Fom simplesymmetry considerations. For a sphere all the dhetions are equivalent,and the tbrœ demaaetieg factors must be equal. Therefcre, D. = Du =Da = j, because the trace of the tensor (which ks the sum of these three

BASIC MAGNETOSTATICS l15

numbers) is 1. Substituting ia eqa (6.1-21) lpltdK t,o the sltm e result asin eqn' (6.1.13).7 Simslarly, for an inânite Gliuder, there is no sxlrlzxv ofdiscontkmity along z, so that D> = 0. 'I'he other two factors should be

' equal for a drcular cross-sedion, so that Dx = Dv = 1a, wàicà leMs to therwœrne reult az in eqn (6.1.18).

Substititing mn (6.1.21) in eqn (6.1.7), the magnetostatic self-energyof a llniformly magneb-rM ellipsoid, whose volume is 7, is

(6.1-22)

which is some sort of aisotropy energy- It is the shqpe tmiaoey tem,which was mentioned, but not desned, in section 5.1.4.

Two particular rxq- are of spedal interest, and botà are elhpsoids forwhich two aN:% are equal. One is the cmse a = b and c > c twhic,h Ls a kiadof egg-shaped partide). It is called a prolate glhur/ïd. The other is shapedmore or less b'ke a disk, or rather a tdying saucer', aad is the cmse a < band b = c. It is called an oblcte cAemïtf. The spàere, c = b = c, is tke Brnn'tof botb-

If tw'o axes are equal, the related two demagnetieg fadors aae thesame. Thus, for a prolate sphezoid N. = Nu, and for an obla* sphearoidN = Nz. ln the cmse of a prolate spàœoid, eqn (6.1.22) emznl Werdore: bebwmtten as

1 - r, a r J 1 , yv ; lrya o jaoum j (6..,, zzaA,Su = -V gkNatMx + .$f: ) + .5 z wWz 1 = j- ;/' (-?$ z - - a x z a2

because AJ.2 + pz: + AJ3 = M.2, wlticà is a constaut Tbis shape anisotropyF ..

energy term h>' the same mathematical fozm as tàe Ast-order uninavixlanisotropy term of section 5.$.1, even though the physical origins aredferent. A similar expression obdously applies lo the cmse of the oblatespheroid, and in dther cmse this shape Misotropy may eodst beside.s .a 11nl'-

,m'n.1 or cubic maaeto ' e Misotropy term, discussed in sedion 5.1.Moreovem the easy a='K of the cystnlllne anisotropy term Ls not necessx.mllyparallel, or related ia any other way, to the easy ars of the shape anisotropyterm, and in pedple there can be aay augle betw. - them.

The ori#nal Stoner-Wohlfarth model, dismxne-' in sedion 5-4, aVumed(2345 no crystalll'n e anisokopy, and dealt only with tàe shape anisotropy ofellipsoids. At &st both prolate and obla* spheroids were considered) butlater extensions addrerxsed mostly prolate sphezoids, wlzich are (2401 verycommon in permxneat magnet ma*rials. Comparing eqn (6.1.23) wiG the61.* term of eqn (5.4.30) it is semz that for a pzolate spheroid all the algebraof section 5.4 is unchanged, but eqn (5.4.32) should be replud by

h, = . (6.1.p,4)(Xz - N.)Ms

116

The Stoner-Wo%lfx-h tlzoz.y is nlmn appncable for pazticles whick haveWth. a cryeltlllne and a shape esotropy. For mwmple, it llaci ven used(241) for a unixvlxl nnlsotropy with botb Tfz and .A-2 superimposed on theskape aaisotropy. Such calculations usnnlly aume that the easy A=-K is ihesame for botk Ansqotropies. If there is a ceztain angle betw- these twoaxej the problem becomœ a little more complicated, but not prohibidvelySO.

The demagneiiziag factors in a general ellipsoid depend only on theratios c/b and s/c, and not on the palues of a, b and c. 'nns, two ellipsoidswkich have the >me shape, bnt a. dxerent volume, have the same N orD. Analytic expressions, albeit in terms of exiptic integralw are lçnown forthe functional fnrm of the dependence (f the demaaetizing factors oa theA:va'r2 raêos. Formulaea raphs, and tables were publishcd (242) by Osbornaad them more ubles were computed (2431 witk a more modern computerprogram. J.n the particular case of a prolate sphezoidj the %pressions forthe demagnetizing fadors are composed from more e-lemen- fundions.Specifcally, by using the notation

P = c/l (> 1), # = 19 - 1 lp, (6.1.25)the demagxetizhg lctors for a prolate spiteroid become

ANOTHER RNERGY TERM

1 1 1 + (' 1 - DzDz = . j- ln - 1 , D= = -

. (6.1.26)# - 1 ( 1 - f 2

For a small f, a powe,r seriœ ex-pansion of the logarithm ln tkis equationleads to

x â1 1 , 1 # - 1D. = =- w + X x + a .

s-- . (6.1.27)Sr- E' k=z

J.n tMs form it is clear that in the limit g -> 1, which ks a sphere, Da = j tas condqded in the foregoing.

Demagaesn'ngfelds and demagncdzhg factors are also defned for non-ellipsoiv bodies. These defnitions will be givo in section 6.3, after theintroductioa of the magnetic charge.

6-2 OriG of DomxlmsThe stage ks now Rt for a demo>tration that it is the maguetostatic self-energy term which is responsible fœ the esteace of the magnetic domainsof secdon 4.1) or at least tkat this enerr term çrelers the stbdivision ofa ferromaaetic crystal into domains. 1 choose a pardcnlxrly simple Am-

ple, wkicâ also proddes a nice example of one of the methods of solvingpotential pzoblems. It is a somewllz.i modl'4ed version of a N:el calculationof an infnite circalar cylinder wlzic.h is subdivided into two domains,

ORIGD OF DOMMNS

R

>z

ll7

FIG. 6.1. A cross-section of an izdllite ecular cylinde subdivided intotwo antîparallel domains.

Tbis cylhder is asslxrned to be magnels'zzazl alongthe rows i!z the coss-section plotted itt Fig. 6.1, which mennq that the magnedzadon vector is

+1 if y > 0, ï.e. 0 K $ :; xMs = Ma = 0, Mr = AJk x

-1 if y < 0, ï.e. g' K $ K 2x(6.2.28)

where y = pe 4 in cylindrical coordinates, with () s $ K 2r. The stepfunction rltn be erpressed by its wemHown Fourier erpansion ,

4 x-.* sialtzn + 1 ) /)M . = -x Ms .aw z a o y ,x=0

(6.2.29)

whîclt mxkM the norrnxl commnent (see Fig. 6.1)2 * sin((2n + 2)4) + sintzn#)Mn = Mp = Mm cos $ = -Ms E (6.2.30).z' 2/. + 1=0

wllc.:n the product oî t:e cosine and sîne functions îs converted into a sumof two sine functions. The sum in this equation rltn now be broken downinto two sums, one with singtzlz + 2)4) and o=e with sin(2n4). H the frstsum the summation index is changed 1om n to n - 1, and ia the secondsum the summation c-an stat't 9om n = 1, because the tlrm with n = 0 iszero anyway- Combining again into one snm, .

2 1 1M= = -Ms S zs - : + sj + 1 sin(2n4).W n=1(6.2.31)

118 ANOTRRR ENERGY TEPM

Adding together the two fzRtions, and sabstituting in mn (6.1.6), one ofthe béuadar.v condidons is

oUQ duout u . -8 u E'=' n sin(2x/l

- = % a % s 2 + 1)(2a - 1) ' (6.2.32)ap dp p=a = a-z ( p,

where R is tàe radias of the cylinder. It is therefore natural to look for asolutoa of the form

m

V = JR IGIP) Sin(2R/), (6.2-33)a=1

where 'ua aze fanctions which have to be de-rrnin ed. Note that bolauseof the uaiqueness of the solution, aay g'uv about the fancdonal form islegitimate if it eventually leads to a Atnction which 6.16 ls all the diferential '

equations and boundary conditions.H the present caze V . M = 0 and eqns (6.1.4) and (6.1.5) are botà

T2U = 0. Substituting 1om eqn (6.2.33)1 we obtain

* d2 1 tf 4x2v2/ = E sà(2z#) m + -

y- -

s w&(p) = 0.p p p*1

Obviously, b0th the diferential eqaatons and the zegulazity at infnit.y arefulfmed by the Rnctions

(p/A)2'è if p s JL%a(p) = c,v X (6-2.35)

(2/J')2'& if p k R

(6.2.34)

wheze e,a are constaats. Moreover, al1 these functions are contiauous atp = .& Substituting th- 'fza in eqa (6.2.33), and substituting the latter ineqn (6.2.32), it is s- that dl the requiremGts av satisfed for the choice

2'ysM4RGz = ' '

.

x(2% + hjt-bl - 1) (6-2-3 6)

lt has thus been shown that the potential inside the cylinder is

2-/s - sintax/) .e. c-5- -

cr hM- 57 (ca + 1)(ax - 1) (a) ' (6-2.3-4

a=l

The 'tdemagnetizing) feld inside this two-domain cylinder is given by mn(6-1.2), and in particular its z-component is

(VD,.n t'y s'Y # t'gSsr. = - -ov = - OS /- - -ay Uin . (6.2.38)Pp p

OWGU OF DOMUS

Substituting U 1om eqn (6.2.37) and performing the dxcentiations,

4% =' ,:sia((2p. - 1)/J g zr.-,x... - -

x .M- E (2,, + 1)(a,, - :.) (s) .

=1(6.2.39)

Intlzis ca-se there is also a p-component, butitdoes not enterthe calculationof the enerr, because it is maldplied by Mu which is zero. According toeqn (6.1.7) the magnctostatic energy pc unit length aloag z is in this cat;e

1 ' 1 2* XSu = -g M%H. dS = -- MxEùpdpdh. (6.2.40)

a 2 0 n

Substituting for Mz from eqn (6.2.28) aud for Jo 1om eqn (6.2.39), aad-

g out the iategration over p,

sg ,vw <= f 2*

su = '-JPMJ V z zyj - j; jc sinltzn - 1)4) :(/) #,? , (2n + 1) (R=(6.2.41)

with+1 10 < 4 s Tr

*@) = (6.2-42)-1 if 'm K 4 K 2?.

If this inteval is separated into integration over the regions / S ';c and4 k 'lrt the integradon is elementars leadlng to

:>. z , 80,su - s.

a Ms E (z,, + y)u(as - gza-=1

* l 1 :x. c z= 2.a2.,/'2 N - = R Ms . (6.2.43).l( * (2n - 1)2 (2s + 1)2 g.>1 '

TNe reuer should not be so naive ms to j'lrnp to the condusion thatall potential problems have suG a nice, aaalytic solution. Obviously, oalysuch e--tKe are chosenfor demonstzadon here, but there are others. Actually,tMs case of an ininite cnrlinde,r is used here only because it %nA this simple(or relatively simple) soludon. The problem of a :pAere subdidded in away slmila.r to Fig. 6.1 %xK also beea solvv (2m1, but oaly by a morecomplu tenbnique which is beyond the scope of this book. At any rate,the importat condusion is obtnsned by comparing tMs result with eqn(6-1.19), nxmely

ne tiorn rk', n 2fa1 ruw -- > 1. (6.2.44)g two cloluaktira.z 4M

l20

Note that tbis resit does not depead on R or Ms. Tkerdore, for any fe-romagnetic material, of aay size, tke magnetostatic energy term b rednadby subdividing the czysul 1.at0 at least two domains.

It is not dîocttlt to extend this calculatîon to more than two domains,and see that further subdivision caa reduce further tàe menetostadc *1#-enerpu And in ca'use 'ëhA's e-xample d a c'yMder may seem to some readersto be a unique mqse, a qualtaténe but convincing argttment will be give,nia the next section, showing that tkis cmse is quite general and tkat thema>etostatic e'nergy prders a subdivkion into domains in any geometry.However) Just because one energy term prefers thjs co ation does notnecessarily me= that it e>n always have its way. There aze otker enerrterms which mttst be considered.

As far as the Misotropy enerr is concprned, there is no diference be-t'ween a nm-fnrm magnetization and the two domains shou in. Fig. 6.1,because if z is an easy es, so ks -z. The anisotropy will only dictatethat z is parallel to a particular mystallographic direction, and is not justany dîrection within tke cylinder for either the uniform magneeation ortke maaetizadon in :$.$m% domain- However, the exckange energy in a fer-romaRet prefers ndgkbours to have parzlle.l spins, and in Fig. 6.1 tkereis a whole snrfnzte for whic.k tke neipbouring spsms on each side ()f it areantiparallel to each other. Therefore, ia order to crcte ths Gmfguradonwork hms to be done against exckange: .and even a very ro'ugh estimationshows that this 1- of exchange enera is much larger than the gain inthe magnetostaiic enerr. The totd energy of the confguration of Fir. 6.1,if tàken =cc'àlp asêin the foregoing calculation, is larger than that of theuniform magnetization, and the physical syste,m will pzefe the latter cmse.

ANOTHER ENERGY TERM

6.2.1 Dopz/in GcJlStill, it takes only a slig;t modiGcahon of the foregoing picture to changetke argumeat. Tke môln point is that the magaetostxuc forca are 'vezy longranged. They control the behaviour over hrge dâstanccs, aud do not chaageconsiderably if a distance of several hundred unst cells ks inserted betweenihe two domains of Fig. 6-1. lt is very deerent from the e-vbxnge, whicllis a ye.zy short-unge force. ït skould be quite clear from chaptez's 2 and 3that ltGects nearest, or maybe next-nearet, neilbours only. It is a ve.rystrong forœ betwœn suc,h nekkbom-s, but it doœ not cxtend to spins whichare much farther away. With small angles betw-n neighbouriqg spins, largechauges of the aagle over a dse-q.nce of many atoms do not involve a laz'geuchange energy- Therefore, the loss in the e-xchange energk rnxn be verymuc,h reduced, if the picture of Fig. 6.1 is approfmately mahtained, buta mall is introduced, in whic,h tke direction of the maretization vectorcxanges gradnally, instead of an abmzpt jump of tke magnetization 1om$ = 0 to $ = x. A more complete tkeoretical treatment will be #ve,n in tkenex't chapter, but in the meantime t:e maia features cau be understood

ORIGIN OF DOMMNS

from a simple, semi-quantitative estimation.When the spin operators are approtmated by classical vectol's, as in

chapter 2, the exchange enerar is as in eqn (2.2.25). And if J ks non-zerobetween aearest neighbours only,

lzu = -)7 Jysï . sj =z -lsz J-I cos $.,t, (6.2.45)

U nekghbours

where 4qj Ls the angle between S2 and Ss A one-dimensional structure isconsidered, in which planes with n spins in each interact with neighbouringplanes. The interaction of plane f is taken only with that at ï + 1 and notwith the other neighbour at f - 1, and a factor 2 is introduced instead, msin the transition to eqn (2.2.26). Then the energy loss fzom the state inwhich a1l spins are aligned is

1 2 ?DC. = 2J5'2aV (1 - cost/yyyj = 4J52a V sin2 $z,g rs JS P.V jh,;,ii ç i(6.2.46)

for small angles. Let this calculation be applied now to the case wherethe direction of the spins changes from 4 = 0 to zr ove,r N such planes.The angle change between planes need not be the same for all planes, anda better scheme will be given in the next chapter, but for simplidty thisangle f,s taken here to be the same. It means that in order to obtain a totalchange of .lr aher N such angles, 4$,.j = 'm/N and the enera loss is

r 2 J52ys.m2t'sex = Jszn E (p') =

x . (6.2.47)

The exchange enerr loss over this wall is, thus, N times smazler than thatof one jump from $ = O to $ = A'. Ob<ously, if this N is suëdently large,the loss in exchange energy can be small enough to be compensated bythe gain in the maaetostatic energy, which makes the subdivision intodomains energeticazy favourable.

Ia prindple, if the Jump from one domain to the other is not abrupt,the calculation of the magnetostatic enerr in the foregoing should alsobe modifed, but the correction is rather small if the wall is not too thick.Eowever, there is another complication in that the anisotropy energy nowenters as well. Two antiparallel domains ctm be along an emsy axis of theMisotropy enera, namely along one of the directions for which this energyterm is a minimum. The spins in the wall, however, must turu out of anemsy direction, so that aaisotropy energy has to be used in creating such awall. Qnlditatively, the anisotropyenerl tries to enforce a thin wall, whilethe exchange energy tries to enforce a thick wall. The above-mentionednumber of planes, N, is therefore determined by minimizing the sum of

122

exchaage and aaisotropy ener#es.The inedtable condusion from all these semi-qualitative arguments is

that none of these three enera terms (eexchange, Ysotzopy and magne-tcstatic) en.n be negleted. Siace domes are aa experimental fact in suf-Nciently' large crystals (a xction 4.1), any reaDsdc calculation of a bulkferromagnet must contzain all these thrœ eneN terms. The foundatlons ofsuch a theory will lx givem in n%xpter 7. Eowever, llefoz.e going into math-ematical detns3n, it is very hstructive to continue a little more with thesemi-qualitative discussion aad œtablish more dearly the pltysial pkctureof a ferromagnet, and the nature of the forces governing its behaviour.

ANOTEER ENERGY TERM

6.2.2 Long cld zS%vt RqngeEvery undergraduate nowadays studiœ Mzxwell's equations, aad most ofthem c-qzn quote the fact tkat the electric and maaetic forces are IOVrangeg becuse the potential decreas% with distance as l/r, whic,h is cslow decrease. Howeve, theere are relatively few, even among profesdonalsworkng on the theory of magnetism, who actually try to underst=d whattMs statement means.

Consider the simple case of a uniformly magnetized empsoid. The feldwhich is meuured at a point hside thts Gpsoid is #ven by eqn (6-1.21),where D is determlned by the mtïo: of its axe. The absolu* size docnot enter. Suppose that this ellipsoid is iHated in such a way that itsSize incease:, but its shape is held the same, ï.e- its axial ratios are keptGmstant- Then the feld is still the Mme ms it waz for the small ellipsoid,which is a function of the axial ratios only. If this indation conGues, evenin the limit of the dlipsoid extending to l-ndnits the demagnetMng Eeld init still depends on the Ga.1 ratios of the anrsace, which is now aa infnitedistaace away- Thus, the Glong raqge'' in the pruent context meaus thatthis range actually extends all the way to l'n6osty.

The obvious conclusion is thn.t in f=omagnetism there is no physicalmeaning to the limit of an :'ncnx' te cryst/ mitho'u.t c snrfaœ. lt 5s not justthû tenbnl' cal pzoblem that in6nite crystals eMnot be made in renlity. Thistechnicality does not cause any docnzlty in o*r felds of physiœ, wherethe assumpdon of inonlty can be l-q.lren as the limit of a crysti whichis large compazed with some sort of a meuuze for the properties underdiscussion. 1n this =e, even in the tkeoretical limit of the custal actuallytending .to -'nGnl-t, the shane of its suece still determine at lemst partof the magnetostatic Oera term, and surXe eects cannot be avoided.Therefore, all calcuhtions of the types decribed in chapters 3 aad 4, whic,hiaore the surfRe by saying that the czystal iq infnite, introduce an error.

This error would not be importltnt if the whole magnetostatic energy .-.

teerm wms z'athe.r small. But it is not- It ts only too often poiated out thatthe uehltnge energy densçty is orders of magnitude lrger thaa the mag-netostatic emergy denray. However, the phydcal systmm is governed by the

ORIGN OF DOMUS 123

tofd.l enera and not by its density. As svted several times in the foregoing,the exeange force has a very short range. It acts essentially only betweenneighbouring atoms, so that its eFective range Ls of the order of the unitcell of the crjrstal. Thereforw the total exchange energy Ls of the order of itxsdensity integrated ovez the volume of a uztit cell, The magnetostatic energydensity is small, but having a long range, it is integrated over the wholevolume of the crystal. For a suldently large crystal, which contains verymany nnn't cells, the total magnetostatic energy is much larger than thetotal exchange enera. It is not negli#ble. On the contre, the exchangeenera controls only the microscopic properties, as in the inside of the do-main wall, but it is the maaetostatic energy term which mostly determinesthe structure of the magnetization distribution over most of the crystal.

lt must be emphasized again that a large error is introduced not oniywhen the magnetostatic energy Ls neglected altogetherr ms it is ia most ofthe calculations described in chaptezs 3 an.d 4. Sometimes a certain ap-proimation for the dipole interaction (76, 2.45) Ls included in the spin wavetheory of Ms ns. temperature. And the review (671 of the renormnlsg-ationgroup calculations cites several cases into which such interactions have beenintroduced. However, aJJ these cases mssume an infnite crystal without asurface, which is inadequate. In. ferromagnetism there is always a surface,even for an in6n'l te cnrstal, and it is the suzface which is Vsponsible for thesubdivision into domains.

'

A11 the calculations in this section were for cmses in which the onlycontribution to the maaetostatic energy term is 6om the discontinuity ofthe derimtive (which depends only on the shape of the surface) becausea11 the mvnmples were thosen to be such that V . M = 0. lf this term isnot zero, the solution of eqn (6.1.4) may be very diFerent h'om the casesin this section, as will be further discussed in the next section. Eowever, itc'an already be stated here that although these other c'ares ex-ist, they aremuch 1- cornrnon in real life thxn those for whic,h V - M = 0. The zeasonis that the main tendency of the maaetostatic energy term.is to subdividelarge crystals into domaizts) in each of which V . M = 0. And even when itis not so in the 1:m11st only a smallpart of the spins are in the walls, so thatthey have a small esed on the overall properties of the crystal. Therefore)a theory which introduces the dipolar interaction but leaves out the surhcetreats a less important term while neglectig the more important one.

However, omitting this largest energy tezm is not always ms bad as itmay sound..paradofcally) the magnetostatic energy term may often beneglected beœ'tise it is the largest energy term. The point is that domn.-tnsare arranged to minirnizm the magnetostatic energy, being the largest term,with very little efed from the exckange, aud only a minor modifcation bythe nmssotropy. For large crystals one may even do quite w6.11g246, 247) by ne-glecting the exchange altogether. But at the minimum, Cu usually reMhesa small value, which may often tuz:a out to be much smaller than the other

124

enerpr tezms; so much so that it Ls ofœn possible 1M6, M8, M9, Zsoi *approlnmate the emerr minîmization by a convnradon for whick E'M = 0.But this ene-rr is only small at the mimjmum aad a deviation 6om tkatconfguratior cAn cost a large amount of maguetœtatfc enera. Therefore,when ciculating the enera of the com'et magnetizadon distzibution, Eumay oen. be negleco, but if it is neglected c pricd, a wrong magnetiza-tion distributioa is reached, wàic,h has a vezy large Eu term.

Because of this property) and because MC.II domain is homogeneouslymagndigM, it is ofl-n possible to get away without the Eu term and withthewrongassnmpdon that the whole czystal is homogeneously maRetized.As explained ilï section 4.1, it is posdble to calculate Ms(T) as if the do-mnlnq did not esq and it worW. But one must always bear in mind that itis wrong in prindplez aad that it works only with some trickq and only forEmited applications. lt is drgerous ground to step on, aad it ks neessaryt,o remembe,r this fad and cikcrlc pxnln case for comNtibility with the as-sumpdon of no domm-nK. Thks aumption can never be taken for gr=ted,and one should certainly not try to extend it beyond its natural rlmits ofvalidity, where a dllenmt thKry is required. For evxrnple, adding a non-zero maoetic Zeld destroys this theoz'y, see section 4.6. This approach hasnever bœn used for the cakulatios oî crltical ecponents, and it is not clearat all whether neglcting the domain structure does or does not have alarge efec't on 't'àese Yculations for any speec case.

This distinction between a Iong-range and a short-rimge force ks alreadysuEcient to rlolva (at least qualitatively) the diëculty which I have ldtopen in cxapte,r 1'- The ovnhnnge force in iron is of the order of 106 Oe, butit takc an application of am extra fdd of about of 103 Oe to wipe out thedomains; aad even a really negligible îeld of 1 Oe rAn make a large (11f-ference to the domain structure. Wlly cannot the $06 Oe îeld accompishwhat a muck smaller f e1d c=? The auswer is that tNe very lvge exn%angeqeld has a ve,zy short range and only enforces small anglu Ytw-n ndgh-bouring spins- lt is not capable of preventing subdivisions into domainsove.r a long rauge. When a magnetîc feld is applied, it does ncf do workagalnst cxchange forces. It works agest magaetostatic forces twhic.h areof this order of 103 Oe in Fe) in rnrnoviag or rearransng domes. And itclm Rcomplish it because it is applied over tke wkole crystak and not onlybetween neighbours.

It shoald be espvinlly noted that the argument about a long and ashort range applie only $o lvge crystals, which eontain a suëciently largenumber of nm't cells. In small particle the long range of Eu dou not makea dferoœ, because tNe integration is only cvried oMc the Bmsted sizeof the cr-ystal. In spite of the elnz-ms of some tkeorists that they omit themagnetostatic enerr term because the crystal is very large alld its surfaceis far away, it adually worW for the opposïte extreme. lt is in swzcll particlesthat the e-xchange is suëdently strong to enforce a uniform maNetization

ANOTHER ENRRGY TERM

MAGNETIC CHARGB

over the whole crystaà although auisotropy also plays a role, as shown inchapte.r 5. lf such a particle is a spitere, the magnetostatic enerr does notente.r at all. If it is an elongated ellipsoid, the magnetostatic eaerr plays

.. only the role of a shape anisotropy, which adds to the other anisotropyterms. Ja dther cmse, the e-xnhxnge force in these small pazticles is toostrong to allow subdivision into domains, or other features of the lazgeparticle. A table of typical nnmerical mlues of these energy terms ca.n befound, for examplq in (251). lt is, thus, in the small, not the large, crystakswhere one should look for a possible validit.y of the spin wa=theory and thecritical e-xponeats. But then, superimposhg the mssnmption of a.n l-nmm-tesample cannot be a very good approvimation for ver.g smaz) particles.

Suëd%tly s'mall partkle are, thereforev homogeneously ma>etizedin z&o applicd feld. 'Phey are then catled in the literature Mngle-domaénparticle. Calculatlg the enc't size at wîich a multi-domain particle turnsIt.O bebzg a single-domlu'n one is not a simple problem, and wi11 be furtherdiscussed în a later Gapter. At this sta& it will only be remarked thatsemi-qualitative estimations of the enerr of the domninK and the wazbetweea them, as done in this sectioaj are all right for rather large particla,for which the accuracy is less importaat. Near the traasition, the energybnln.nce îs rather delicate, aAd a higher accuracy is needed, eve.n thoughthis point was ignored in esl-mations published in the 1940s and 195Os.The frst rigoroms calculation (252) considered a sphere sliced into planes,as rourhly done in sedion 6.2.1 here, but calculated uactly the exchange:nm'sotropy, azld maretostatic energes for thnr.n glice. lt rpnrled the valueof 37 nm for the radius %low whic.h a cobalt ohere is a single domain,and above which it should divide tto'two domains. Bve,n this calculatîontcrned out to be iaaccurate, because it was later found (253) that the totaleneror in a sphere can be further reduœd by meng tNe domains cuzved,with a rlindrical symmetrp This modifcation reduced the lcritical radius'for being a single domain to 34n= in cobalt.

6.3. Magnetic ChnrgeUndergraduate textbooks give a formal soluuon to thedifereptixl equationsand boundav conditio'ns in eqns (6.1.4)-(6.1.6), which ca,n be etten inthe form

V' - :u(r') n . M%'%rJ(r) = .:*- - dm' + î l ds? (6.3.48)4x lr - r/ I lr - v' l '

where V' contains derivatives with respect to tîe components of rJ, thefrst integri is over the fromagnetic bodies, the second integral is overtheir sueces, and :rt is the outward normal.

This solution does not solve the problem ia the sense that mz rxn forgetabout the df erential equations. lt is Ch;G easier to solve the diferemtial

126

muations, as in the evnmples Xven in the previous section, thamto carzyout the ineradons in this equadom As a nice zlustration) the reader maytzy to obtain the solution of eqn (6.1.11) for a homogeneusly maaeties'phere by carrying out tke in+grations ia mn (6.3.4E). It is certainly pos-sible to do it, because tke two mressioc are rnxthematically idotical.But the întegration is denitely not trlvial. Actually, more oftea tkan notlaualytic intevation from tkis solution is rathc mzmbersome and not emsyto perform, unless some traasformation is frst applied to ât the particularcmse- It is not very 'IK-GII for mlmerical integration eitheer, except for certainspedal cmses, because most of the contribution to the integr=d is usuazly1om the vidnity of the singulim-ty at F = r, where it is not emsy to attean adequate acccas'y. Msq tke frst term in eqn (6.3.V) is aa integrationover the volume, which is a tkree-fold întegration. In order to calttnlxte theenergy, the w<mlt hM to be substitutM in eqn (6.1.2), and tken substitutedin eqn (6.1.7), witic.h involvœ anoier tkree-fold intevation. lt may iangein the near future: but right now a six-fold numerkal integration to any de-cent accuracy is beyond tke ca-bility of M-Kting computers) even tkougll,tlme six-fold htegrations of this sort (254) Mn6 bœn cvried out.

Tkis formxl solution is more uspfnl whe.n $he srst integral vanishes,and there is only the second one with a twmfold integration. The energycalculatîon then involvœ only a four-fold integratp and if one or t'wo of theseintecations eltn be exm-ed out xnnlytirltllyj tke numerical problem becomesquîte mxnageable. But the mœt importaat application of eqn (6.3.48) isbued on its qnalitaténe propertiœ: which allow an insight into what themagnetostatic enerr prefers Mthout actually doing auy calculations. Thîspossibility of udng phyïcal ttuition is due to the formal Iorrn of tkeintegrals ia eqn (6.3.48), wlkic.à contain 1/r. This hctoralso appears în theelectrostatic potential of a point chcrge, which allows the llrnt integral tobe intezweted as the potential due to a spatial distzibutîon of a rolwmecharge, with a charge density -V . M. Sl-ma-larly, tke sexnd integral canbe ccsidered ms if it was Gpressîng the potential due to a suviat.e cllorgewbose surflme density is M .n. Of course, these càarges do not Hst. Ma.nyboob ek-plaân that the diFerence betwen electrostatic and maaetostadcsis that there is no ma>etic chrgeo and that these integrals ikave only amathematical meaningj and do not express any pkydc.al reality. However,it is neve.r necessvy for any nseW mathematical tool to kave a physicalmeaning. There is no mnl physical ckargeo but tke mathematical identîtylxztwœn thœe integrals and those whic.à involve a charge makes ît possibleto use the Howledgeabout a real cbnzge to guess tke quazit,ative propehiesof the magnetostatic potential.

Ia partlcular, we know that sirnl-lar càakrges zepel pxzth other- Therdoreja volume Otribution of such a chvge r-qJn be susteed only if it is heldby other forces. Lelt to itself, the chargœ anmkere in the volume willrepel epm.b othe as far as they eztn, wbich is a11 the way to the surface.

ANOTEER ENERGY TERM

MAGNETIC CRAR GE 127

++++ ++--

- +- +- +- +- + l t t

--++

(a.) (b) (c)FIG. 6.2. Sckematic represotation of the surface charge in a particle mag-

netized along the long and the short a'dsl and the Mme particle subdi-vided into two antiparaltel donosns. ,'

Therefore, the maaetostatic enera term by i/elf will prde to avoid thevolume charge altogether and create only a domain strudure with a chnrgeon the outer surface. If there is ny volnme charge at a11, it can only comeout of a compromise with another enera term, e-g. within certain trpes ofthe wall between domes. Because the maaetostatic term is usually thelargœt force in seciently large c , and most of the magnetizationstructure is ananged to ft th% tarm, a volume charge will baardly evu beencountered. However, it must always be bon'rp in mind that if a stzucturewhich involves a volume c'harge îs introduced into a (wtain calnnlxtionjthe maaetostadc energy due to this volume cAarge is sof negb-rible. It iscften convenient to introduce such a structu're ia œrtain problems, and itis uezy tempting then to forget about the volume charge and arxsume thatit probably doœ not have a large eect. It i% therefore, necessary to wartage that if the volnme Garge doœ not enter other calculations, it Ls notbecause it is negli#bly small, but because it is Ntremely large. It is onlyby avoid-lng ît that the magnetœtatic emera can be minimized to a smallvalue, and if this càaxge is allowed to creep în, Su e-xn increase enormously.

A slmilnr argttment applies to the surface charge as well. Consider, forevnmple, the exw-q shown ia Fig. 6.2. Tbe single-domain structure in (a) hmsthe O'rne rhltege densit as the one in (b). But in (a) this charge is spreGoveralargerareathan in (b). nerefore, the enera of the case (b) is smatlerthan that of case (a). For Gpsoids, this rctllt can also be obtained from

128

the Hown analydc solution, according to which the demagnetizing Mtoris smallest along the longest aHs, and tendsng to zero in the limit of aninfnite cylinder. Howevez, the concluhon from Fig. 6.2 is easier to see, andit also applies to otber bodies, and not only to ellipsoids.

J.n case (c) the total surhce curge is the sxme as in cmse (b). However,the subdivision into two domm-ns malces some of the negafve charge fromthe bottom sudace move to the top, replaclg part of the positive chargethere that is moved to the bottom. Since unlike charles attract p-qmb oiher,the structure in (c) is more favourable) and its eaera ls smaller than that ofthe strudure in (b). Again, this conclusion obviously appliœ to any shape ofthe magnetic pazticle, and not only to the eltipsoid shown seematically inFig. 6.2. Akso, the snme argument applies to further subdivision into morcthan two domains. It may thus be concluded tkat the magnetostatic energyterm prcfers a domain covguration over a uniform magnetizathon for anyferromagnetic body, and tàat it would rather coneue th'-q subdividonindefnitely, unles stopped by the competition with the other enera terms.Therefore, a uniform maaetization e.stn only est either in suEdentlysmall particlc, or in a ctystal to which a suEciently Iarge magnetic feldis applied. A large magnedc feld can wipe out the domains and rotata themavetizadon to it,s own direction. '

Before condudhg this discusdon of eqn (6.3.V) it. will be remarked forthe sake of completeness that this formal solution is nlnn usel fortwo otherpurposes, even though these are usually Ested in undergraduate tebooks.One ks that dn (6.3.48) ks an pvlKteace tkeorem. It proves t'hat tkere is atleast one solutioù to the set of eqns (6.1.4) to (6.1.6), thus completing tkeproof given in section 6.1 that 'iltiq Rt of equations cannot kave morc thanone solutiom The second remark is that eqn (6-3.48) proves the principleof sneoosition. Since everything is linear in those integrals, it is alwaysposdble to calculate separately, even by dif-nt methods, the poteltialcreatd by diferent parts of the charge, and then add them togethen

ANOTHER ENERGY TERM

6.3.1 Gene, DernagnedzctïonA cmse of spedal practical interest ts a homogeneously magnetized body.The domain covguration in zero, or small, appnH feld is ver.y complictedand vezy rliecult to reproduce. It varies fzom one umple to another, andeven for the same Kstmple it depends on the Mstoc of the applied Eeld(see Fig. 4.1)- In order to ezzibrate meuurements, one must stxaz't withsomet%lng which eztn be related to the matezial, and not to any spexcsample. The best case is a sample in a suldently large feld, for which onecan at least hope that the magnetization is held paratlel (orxlmost parallel)fo tke direction of tlze feld throughout Ge whole umple- However, the feldinside a ferromagnet is not the same as the feld out,sideo and the d'-Ference(called the dema>etizing' feld) is a Gnction of the shn6 of the sample. Ar=onable estimate of this demagnetization must be kubtracted in order to

MAGNEXC CHARGE l29

remove the efects of the particular sample and renrrb the intrinsic properties()f the matedal. Therefore, the de6nition of this feld, given only for anellipsoid in the previous section, is extended here to othe.r bodia.

la a uniformly magnetized bodyg V . M = 0, and the ftrs't integralizt eqn (6.3.48) mnishes. Substituting the second term in eqn (6.1.2), the(demaaetizing) feld inside thé ferromagnetic materiaz is

(6.3.49)

where M was lmlcen out of the inteaal, because it is assumed to be aconstcnt. For the same reason, M can be moved to the left cî the digeren-tiations, so that eqn (6.3.49) essentially means that tuz!h component of ELs a Iinecr hmction of the components, M., Mv, and Mz- Also, mn (6.1.:)is in this case,

1Sv = -yM . Hd'z; (6.3.50)

where again the constant M is moved in lont of the integral. These tworelations m<o.n that the magnetostatic energy, h this case of a homoge-neously magnethed body, is a quadratic /=n in the components of M. ltrgkn, thereforej be written in kbe fœm

s = - 7Jxv ( M . j j v .n

s g :$ s ' ) ,

(6.3.51)

where Nzz etc. are constants that depend only on the shape of the partide-lt is always possible to rotate the azxes so that tlds quadratic form becomesthe same as eqn (6.1.22). The htter is, thus, the most general fo= ofthe ma>ctostatic energy of a 'unvoonly rrzognefz'zed ferromagnetic body,which applies to auy shape, and not only to ellipsoids- MorKver, by using '

the properiie of the function l/s it exn readizy be shown (1, 255) that i'athe diagonnll'',M form of eqn (6.1.22) all three components N=, Nv and Nzare non-negative numbers, whose sum (whîch is the trace of the tensor .N) is

'

%. Therefore, as far ms the ener& is concerned, any unifozmly magnetizedferromagnedc body behave in the same way as an ellipsoid which has thesame volume. This statement is Hown as the Brown-Morrish theozem.

It should be particuhrly noted that this theorem do% not even reqlzirea simply eonnected body, and appliu even to a body that contes ccdtied.Of couzse, smmetzy considerations may be used just as in the case of andlipsoid. Por example, a cube must have lhree eqnal demagnething factors-Therefore, the demagnetizing lctor of a cube is the same as that of a sphea.ewhich has the same volume, if that cube is nnl'formly magnetized. Eowevervsuch a statement hms nothing to do with the quetioa of whether a cubecan be brought to this state of being uniformly magnetlzed, ahd how to

1 . r , s8v = -7 (x5 kz k5.f2Z + A la x%1'x -l'td'p + . . ,) ,2

130

do it. lt is generally assumed that a suëdently lvge, nniform appliedfeld ca.n bring tlze magnetkzation in the cube to be neazly ulform, but itt-qk'es a special, non-uniform ield to make the cube completely nniformlymaretized.. The mi i.a dference between a.n. ellipsoid and any othe,r body is thatthe demagnetizing seld inside an ellipsoid is nnfonn, namely it is the Mmeat every point inside the ellipsoid, whicN is not tnze for any non-ellipsoidalshape- Although the energy of the latter is the same as that of a certaineEipsoid, tàis energy is an aveage over a certain Eeld distribution. Fornon-ellipsoidal bodies in a large applied feld, Happl, it is still customazy todezne a dœnmetizing hctor, N, and take the interaal f.eld as

ANOTHERENERGY TBRM

Hqo' = mpp: - NM, (6.3.52)because it is the oaly way to eb'msnx+ the efed of the shape of the sam-ple and wuztlk the intrinsic propertiœ of the ma+rial. However, in non-ellimoidal bodiœ it ks only a,a approzmatiop and it #ves only an averageof tke ia-aa.l Eeld. Only in an eltipsoid is the average tke =trne as the feldat evezy point. .

ln pedple, the dema>etizing hctors (namely, the components of thetensor .N) can be calculated by evaluating the potential of .thq sllrfnaceclzarge in eqn (6.3.48) for tke particular gxmetry, substitueg in ecm(6.1.2) to fnd the âeld, and then t ' the appropriate average of thatfeld. Two dferent denltions of averagœ are IISM in pradice. One 11- afeld average ove,r the whole volume of t:e sample, leading to a demagne-tizing hdor which is called the magnetomet6c demagn ' - fRtor. Theother defaitioa is an average ove,r the middle coss-section of the crys-tal perpendictzlaz to the frecdon of the applied îeldj lsuzling to what isknowa as the scllidtic demunetldng hctor. Some of tlt- calcttlations eAnbe carried otzt analydcally, and some call for a numerical eeuation, withor without certaia appremations.

Detltllq of such eeuations and tablœ ok both knds of demagaetidng2factors, can be found in the hteratuze, and are outside the scope of tksbook- .No speec evxmple will be given here, and only several leadiagrderencœ will be mentioned. Tablœ of both demagnetizing hctors in arectangttlar prism Gst for the cmse of L1) one dimension extending to inf-nits and for the cmse of (2ö6) one square coss-section. For a Mite cirfu'ln.rcylinder there are tablœ (256, 257), a long review with formulae, tables andvaphs (258J, aad a sophle-cated computational scheme (2594. There Ls aspccial study of single or double thin ilms (26% 2614 and there are alsosome theorems (26% 263) of a more general nature, and an attempt (264)at a Nrst-order correction f or tke cmse of a slizlztly non-unifoz'm maaeti-zadon distribution- And there is also a detailed discussion (2651 of certaândrawbzmlrq in practical applications. '

UNXS l21

6.4 UnitsOlder tutbooks used tke cgs system of units, ia whic,h the basic units arethe centimetrw gram aad second. Môdezn textbooks for undergraduateàa,'e switc'hed completely to thesystem called S1, for Systènte zhferxafoorlcld'Unçtls, and it eztn safely be assumed that the reader ks more Gml'll-ar withit than with the cgs. lt may thus seem simpler to adopt the SI units forthis book ms well. However, righdy or wzongly, pradically all the modernliterature on magnehsm *-11 uses the so-called Gaussi= cgs estem of upits.And tke reader mu'st become fxvn'linr with it, if only in order to be able tozead al1 this 'pqblished literature.

For Mrne people, converting iX:II Si units has become an obsession,bordering on a rdigious conviction to abozsh heresy aad make everybodyuse the Strue' units. However, there is no way of ignoring the fMt tllatthere are many reseamhers who have not been convez'ted, and it sexs thatthey will not be for m=y years to comû ln any caseo the use of units isonly a matter of conveniuce, oz as Brown (266) phzaEed it: Cdimensions arethe tvention of man, aad man Ls at liberty to aœign them in any way hepleases, as long as he îs coasisteat throughout any one mterrelated set ofcalculations'. nis tuton-nl (266) nlm deoes, and describes the history otsystems of lmsts', au.d it.s reading is lu-rhly recommended. ,

The best source for the demitions of the Gaurvqian ; and itsconversion to SI; is the appendh to the I.U.P.A.P. report (267J on units.1 will only stao briey the impoexnt conversion fadors, in words an.dnot as a table, accordiag to the good advice of Brown (2662: tAt a.ll cœts,avoid conversion tables; with them, you aever know whether to multiply ordivide-' And tàen, for tàe rest of the book,onlythe Gan--nn cgs system willbe used. Evea the Vtor 'ys of Brown wbâch has been used ia this chapteraad i'n section 1-1 will not be ca'ried any farthez. It will be replaced Somthe n%t Gapter by the cgs value of 47r.

The cgs unit of maretic Eeld, Jfj is the xsted (Oe). The S1 unit is 1A/m= 47 x 10-3 Oe. Or 1Oe ra 79.6 A/m. The Oe is the Rœme ms Gb/cm,where the gilbez't (Gb) ks the cgs Alnlt for the magnetic potential, U. Thelatter ks measuxed by the ampere (A) in SI, and the number of A has to bemultiplied by 0.4* to obtaia the number of Gb.

The magnetic iadudion, AIM Hown as the magnetic Qttx density, Bt ismeasured in gaurxs (G) in the egs system. ln this system, H and B havethe same dim<mKions (see mn (1.1.2))j and some years ago H was alsomeasured in gauss. However, ie ttnl'ts now have dxerent nxmes. The SIunit is Wb/mz) also rxlled the tœla (T), aad 1 T = 104 G.

The permeability JI is a dsmpnm-oaless number in the cgs system, and p.zof eq.a (1.1.2) should Just be replMed by the namber 1. ln S1, for whicà eqn(1.1.2) is written, the pezmeability of :1* space is Jzo = 4r x 10-U H/m. Inthis system the relatins permeability: yr = pjgnt is also used, as defned in

132

eqn (1.1.4). The value of Jzr is equal to tltat of the cgs p..The magnetizatiûn, M, sometimes cazed the tloltlrpz magnetization,

ks the dipole moment per unit volume. J.II cgs it is measured in emu, oremu/cm3, even though emu is not really a lznit.in any sense of the word.The number in emu/cmS kas to be multiplied by 1O3 to convezt it to A/m.Oftea 4zrM is specled instead of M, and then it is measured in G azB is, see eqn (1.1.2). J.f M is divided by the density of tke matersxal, it isknown as the mass magnetization) and measured by emu/g in the Gaussiansystem. Jt has the sqme numerical vaiue as A.mz/kg, which îs tke SI unit.The susceptibility and permeability aze dimensionless numbers in the cgssystem) and the permeability of the vacuum Ls numerically 1.

The demagnetization factors D and N are dimensionless botk in cgsand SI, but there is the factor 47 in N as dvned in this chapter by tket'wo values of % . The azisotropy constant K, defned in chapter 5, has tkedimension of an enerpr density, namely energy per unit volume. J.n cgs it ismemsmed by erg/mnz, and in SI the tmit is J/m3, wlzich equals 10 erg/cmZ.A11 the otker conversion factors skould be obvious now, and it is hoped thath d 'on égV re them out.t e rea er .

ANOTHER ENERGY TERM

BASIC MICROMAGNETICS

lt can be concluded fzom the lazt chapter that there is no way to neglect anyone of the three energy terms? exchange, anisotropy, and magnetostatic,and a11 three must be taken into account in any realistic theory of thernnagnetization processu-Et would havebeen nice if the other terms could beadded to the Eeisenberg Hamiltoniaa, at least as a perturbation. But thisHamiltonian cannot even be solved quantum mechanically witho'at theseterms unless quite rough approfmations are introduced. Therefore, untila better theory can be developed, the only way is to fneglect' quantummechanics, ignore the atomic nature of matter, and use classical physics ina continnous medium.

Such aclassicaltheory has been developed in parallelwith the quartum-mechanical studies of aVs(T) which just ignore magnetostatics. 1ts history istold in (2682, 9om the start with a 1:35 paper of Landau and Lifshitz on thestructure of the wall between two antiparallel domains, and several worksof Brown in 1940-1. Brown gave tbis theory the name micromagnetics,because what he had in mind at frst was the study of the detalls of thewalls which separate domnins, a-s distinguished from the domain theory,which considered the domains, but took the walls to be a negligible part ofspace. The name is somewhat misleading, because the microscopic detailsof the atomic structure are ignored, and the material is considered h'om amacroscopic point of view by taldng it to be continuous.

Part of the classical approach is to replace the spjns by classical vectors,which has already been done in chapter 2. But on top of that, a classicalthèory which cxn be used together with Maxwell's equations must have aclassical energy term that cnn replace the quantum-mechanical exchangeinteraction, in the limit of a conïinuo'as material

7-1 lcbssical' ExchangeAs seen in section 6.2.1, the exchange enerpr among spins can be writtenin terms of the aztglez 4:,j between spin 4 and spin j, a-s in eqn (6.2.45). Ashas been' explained there: the angles between neighbours are expected tobe almays small: because the exchange forces are vezy strong over a shortrange, and will not allow any large azigle to develop. For small 14ço.I R ispossible to use the saae approfmation as in the particular case of parallelplanes with w, spins in each, leadâng to eqn (6.2.46/ and write

l34 BASIC MJCROMAGNETICS

K%

S:

Flc. 7.1. Schematic representation of tlle CIIaIN in tlle angle beœeenneighboadng spins ï and j, aud the position vector s: between them. .

6..K = JS2 F) 4? j%,jnighbotuv

%%x%

%

after subtracting-the emergy of the state ia which all spins are aligaed) andwhiG isJ used as a reference state in tùis calculation. It me=s redefningthe zero cf tbe exchange energy, whicll is always legitimatw provided thatit is done consistently-

For smxn ugles, 1401 = lw - mjl, where m k a tznït vector wilicll Lsparallel to tlle local spin direcdon (see Fig. 7-1). Note that this defnitionalso means that m Ls paralle,l to the local diretion of the magnetizationvedor, M) and it is actuallz the same m as in eqn (5-1.6) whenever M isa continuoms ' 1% wlzicll is dvned not only at the lattice points. Forsuch a vadable, the Grsvorder erpansion in a Taylor series is

lmï - mJl = l(s# - V)mI, (7.1-2)wllere ss is tlle posiîon tlecfer pointiag 9om lattice point ï to j (see Fig.7.1). Substituting in eqn (7.1.1),

s = JS2 ((sy . V) m!2 (7.1.3)f 5 (

where the second snrnm ation is over tlle posiuon vectors from lattice pointç to all its neighbours. For example, for a simple cubic lattice with a lat-tice consunt c, this snm Ls over the .W.z vectors sy = c(+1, +l, +1). Thissnzmmation iz readily carried out for a11 thz'e types of cubic lattice, andit is seea that they all lead to the sxme expression, and difer only in amultipûcative factor.

Chan#ng t:e mmrnation over ï to aa integral over the ferromagneticbody, tlle result is that for cubic crystals the exeange enerr is

LCLASSICAL' BXCHANGE

1 , sz z , scxc = -a c tkvzrlz? + r.7', $,?ul + 'vTmz) j ,

135

where2JS2

C = c7a

a is the edge of the unit cell, and c = 1p 2 and 4 for a simple cubic, bccand fcc respecdvely. For a hexagonal close-pafked cystal, such az cobalt,summation over the sç vectors leads to the same result as in eqn (7.1.4)Aonly with

'

4xN2J52C = , (7.1.6)

a.where a is the distance between nearœt neighbours.

For lower symmetries, eqn (7.1.4) has to be somewhat modled. Butfor most cases of any practical interest this equation cnn be Vken as agood approvimation for the exchange energp in as much as the assumptionof a continuous material is a good approimation to physical re/ity. Theconstant C is then Vken as one of the physical parameters of the material,whose value is obtained by ftting the results of the theory to one of the mea-surements. Of course, it ccn be obtained lom the theoretica,l expressionsin eqn (7.1.5) or eqn (7.1.6), whenever the exchange integr'al J is lmown.However, J depends on the temperature, as explained in sectfon 3.5, andthc mlue of J near Tc is not useful for micomagnetics calculations whichare usually applied at, or near, room temperature. The best values for thisezchanne constant C are usually obtained from ferromagnetic resonance.The order of magnitude for b0th Fe and Ni is C ra 2 x 10-6 erg/c.

The factor l in the defnition of C in eqn (7.1.4) is quite arbitrary,2and was introduced by Brown (145) in order to avoid a factor of 2 in thedxerential equations which minimize the energy; ard which will be in-troduced in section 8.3. Many workers prefer to write the energy ia eqn(7.1.4) without the Gactor lz and defne a dxerent constant of the material;A, where C = 2.4. It ofien causes confusion because 30th A and C arereferred to as the fexchange constant of the material' and it is not alwaysdear which of the two ks used in any particular calculation.

The exGange energy of eqn (7.1.4) is avery powerful and useful tool forsolviag problems in which tàe direction of the magnetization vector varieslom one point to another in the crystal. Rs size is assumed to be Ms(T)everywhere, as discussed in. section 4.1. This energy term is zero for thecase of atired magnetization, when all the derivatives vanksh, which is theway its zero has been defned here. It is large for large spatial variations,with large derivatives, witic,h is what one expects the cxchange energy totry to avoid. However, there are certain limitations for the application ofthis energy expression which must be emphashed. As is the case with anytheory, one should never be c-arried away and try to apply this theory

136

beyond the naturi vazidity of its apprnxn'mations. It is therefore importantto specify what these liets are.

ne most olwious restziction is connected with the bmsic assalmption ofa continuous material, which can only be valid as long as any chzuacteristiclength it deals with is ver.g large compared with the size of a unit celt Itis not sometbing whicb eztn be gttarauteed in advance. It is Just necessaryto bear in mind that if any micromagnetics calcdation comes up with apazameter that has a dimension of length, the vult is reliable only if thisquaatity îs mucx lazger thaa the unit cells.

The second, and 1- obvious, nmitation is that tke temperature is nottoo high. In chan#ng over 1om tEe spins at the lattice points to a contkm-ous vadable, M, the mMnitude of tàis vector M comes out automatiœlyms a constaat over the wkole cystal. lt is also an expmm'mental Latt thatthe magnitude of M within the domains Ls a Gmstant of the materiakMs(T): whic,h depends only on the temperatuze, as diseussed in section4.1. Eowever, the picture of fzxu spts at the lattice points is not a goodapprovlmation to real materials, as d-lpztussed in cbapte,r 3, and the experi-mental fnztt that

lMl = Ms(T) (7.1.7)

BASIC MICROMAGNETICS

is only tnze as a.u average over a rather largevolume-lt cnannot be stridly soat every point when there is enough thermat ductuation to make a diferencebetween one point and another. For hc.k of a better model,' tEe theoz'y ofmicromagnetics assqmes that eqn (7.1.?) holds ewer-/zera Therdore, thistheory, as it isy .cazmot be carried all the way to the viclnlty of Tc, whereeven small locd felds may change tàe mavitude of M.

The necœsazy modiications of the theory, Gfore it can be applied tolligh temperatures, are not vezy dear, even though there have been someattempts to geneaollze it. The biggest step ia this direction was that ofMinnaja (269) who showed that in the prœence of thermal fuctuations, theuchaage eaergy density in eqn (7.1.4) should be replaced by

(7.1.8)

where M is the maaitude of the vector M and is a fudion of space.However, Minnaja did not do the next necessary step, which is to replaceeqn (7.1-7) by auother reladon, whicb should be used to detlrmlne tbis M.Msnnaja (269) jMst ignored eqn (7.1.7) and it left him Mt.h too much choiceof possible solutions for the diferential equations, which cannot do for ageneY theorp A true genemlivAtion of micromaNetics should (270) replacemn (7.1-7) by something whic,k tends to it în the limit of 1ow temperatures)and has a better physical mennsng at high temperatures. This part has notlxsen done yet, aad an attxpt to xlve a speel cxase (271) was not verysucceqfnl; and was neve,r eended to other problems. lt was later suggested

C a a a'tpe = ava g(VA&) + (VMv) + (VMa) j ,

ICLASSICAL' EXCMNGE 137

(272) that eqn (C.1.7) be modifed at high temperatures by adding an extraenergy tezm whose density is prchporkional to (lM1 - Ms)2j and this formwas used (272) to solve a certain problem uzzder some approadmations. Jtwa: noted (273) that these approfmations were not really needed for thatsolution, but there wms no furthez development of this idea.

In the cmse of nvclection, whic.h will be discussed in chapter 9, eqn(7.1.7) ca.n actually be ignored, for reasons which will be explained there.Minnaja (269) solved his high-temperature equations for thLs case of nucle-ation (in an infnite plate), whlch is legitimate. Similar nucleation at Mghtemperatures was then calculated (274; for the case of an inf nite cylsnder.

It should also be noted that the approfmation used here is valid only forsmalï cngle.s between neighbouring spsns. Since the exchange is the largestforce over a shozt range, it ca.n be expected that these angles are generlllyvery small indeed. However, this general rule does not exclude some ex-ceptions in unusual cases, such a,s a corner where tke magnetization mustturn around due to some constraints on other energy tnrrrm. Formally, adiscontinuous jump of an angle calls for an in4nite exchange energy, if eqn(7.1.4) is taken to be literally correct. But the point is that it should notbe taken to be literally correct. This equation Ls, after all, only an approx-Hation to eqa (6.2.45), and the lattet has no infnities. Even eqn (7.1.1)is always flnt'te, and approfmating it by somethîng that becomes infniteonly means that the approfmation is not applicable for that pazticularcase, whic.h must be studied by other methods. .

It cmn be argued that an occasional angular jump in some place meansthat a particular palr of spins has a much higher energy than any otherpair in the crystal, which does not seem like an energy znfnfzntlm. Howeverythis argument cannot r'ulc o'at the possibility that this arrangement willbe aa energy minimum under some spedal conditions, and it certainlydoes not Justià taldng the apparent înfnity of the exchange too seriously.This point has often been overlooked (2704 and ledj for ecample, to specialsolutions (275) for a certain tsingular' point in a particular type of a domainwall. That solution rnlm'mt-zes only the exchange energy, because this termtgoes to in4ni ty for'r -+ O proportional to 1/r2, and exceeds a1l other energy .

terms'. Even Brown, who was alwaysverycaœeftzl with his defnitions, madethis mistake, and in a footnote on p. 67 of (145j he ruled out a certainconfguration, because it 'would entail inf nite exchange enerr'.

This problem is a zeal one, for certain special cases, but it has no generalsolution, and a certain attempt (2761 to solve it has essentially failed. Thereîs no altpaative to the use of eqn (7.1.4) for most problems, and somespecial techniques for special problems. It should only be borne in mindthat there are cases for which the generalzule Ols, a'ad should not be used.

ln the summation over the positiön vectors s: that 1ed to eqn (7.1.4),it was implicitly assumed that all of them are inside the crystal. When thelattice point ï is on the suzface, some of these neighbours may be missing,

l38

aad the sum mxy come up dxerent than at intemal lattice points. It is nota serious problemp aad for all pradical purposes it is suKcient to keep eqn(7.1.4) a,s it iq assumiag it applies everywhere, aûd add aaothe.r eaerr termwhich asects the sarface only. Actually, tMs mndi4cation of tNe Gcàaagenear the snrfnr- is only one of several conkibutions (145) to the s'ttrfacetmï.sotrppp enevgy tcmj alzeuy mentioned in section 5.1.5.

The fnrm of eqn (7.1.4) is partscularly stlited oaly to Cmesiaa coordi-nates. It cVs for certaân tmnsformations in cx-forwhlcE other coordiaatesystems are prefœable for auy reasom It is not verjr diëcult to carry outihese trxnKformations, but it is euier if they cxn be avoided altogether.For this purpose it whs suggested (27% that tNe Kxchaage enerc demsityi.x eqa (7.1.4) 1x replaced by

BASIC MICROMAGNETIV

C z zm. =

aw E(V ' M) + (V x M) j , (7.1-9)

because for a vector of %ed magnitude the difezuce betw-n this empr-sion aad tNe one ia eqn (7.1.4) is (277) a divergence of a certG vector. Thevolume integral over the latte.r eltn be traasformed to a snrfn.ce inteaal,by using the divergence t/eorem. Therdorw this diNrence zedefnes onlythe sudace anisokopy term, aad does not cbœnge the exchange enerr inthe bulk, aad in tNe fo= of eqn (7-1.9) it is easier to chaage to a difer-ent coordinate system. This suggetion, however, has never beea ased byanybody else, and will not l)e used here dthem

7.2 The Izudaa and Lifshitz Wall.&s a frst illussration fœ tNe use of this dassical exchange enerr, a bette,rsolution will be give,a heae for the bœt strudure of tNe wall betweea =-tiparallel domains. This wall has already b-n dincussed in section 6.2.1)but very roul appremations w&e used therel which can at best demon-stmte the feasibility of its evlqtence. A much better approach is to rnlnlrnlzethe enerr of tNe problem, using the same appremations whiclz were frstintroduced (268) ia 1935.

For thss purpose, conslder aa insnite crystal, witich has a uniandalxnsnotropy of tNe type of eqn (5.1.7). The domains will arraage themselveswitb their magnethation parallel and antilmmllel to the easy xnsnotropyn.='s which iô defned kere as the z-a='K tsR section 5.1). We defne thez-n='K along tNe dirvson ia which the magnetization chuges from tke-z- to the +z-directionv namezy from mz = -1 to mg = +l, where mis defned by eqn (5.1.6). In the wall betw-n the domains, m tilts out ofthe z-direction, which c,an be either towards z or towards y. Houver, aam. which is a functiim of :r mexnm a non-zero V . M and #ves rise to alarge maRetœtatic eneo contribution. Obdously, the enerr is lower ifmz = 0: and only mu aad mg are 10 to be functions of s. Combiaing

THE LANDAU AND LIFSHXTZ WALL 13S

the anisotropy energy density 1om eqn (5.1.7) with the exchange eneradensity fzom eqn (7.1.4), the total energy density for this case Ls

2 z1 Jrzw dmz'tn = KLmz .1- Kgmî + -C ' +s. . y a (jz (Lz (7.2.10)

The magnetostatic enera is left out, because V . M z'x % so that thereis no volume charge, and the surface charge Ls neglected by the assumptionof an ûinfnite' crystal. The reader has already been warned in section 6.2.2that such an assumption has no physical meaning, and that leaving out thesudace charge by such an argument is never justïed. However, this kaow-ledge nnme much later, and for many ypltrs everybody was convinced thatthis approfmation was 6llly justifedj at least for bulk materials. Actually,there are still many who believe, against strong evidence, that at least theenergy calculated here is a good appro-xn'm ation to the walls in very hrgecrystals. Now it is known that the approfmation Ls not really Ju' stifed,and that the magnetization structure in a wall dou not look at a11 like theone calcalated in this section. Still, this structure is very important from ahistorical point of view, being the frst study in micromagnetics. It is alsoa nice and easy problem to solve) and as such it makes a good întroductionto the more dihcult problems of micromagnetics.

The vector m is a 'uzùif vector, which means that its magnitude is 11and 77z2 +m2 = 1. The easiest way to enforce such a constraint is to defneF zan angle, % by the relation

mz = cos: and mv = sin 0. (7.2.11)Substituting ln eqn (7.2.10)) and integrating the energy density over x, thetotal energy per unit area in the vz-plane is

(7.2.12)

The Euler diFerential equation for miuimizîng this integral Ls

+9 aC a - 2Xz sin 0 coso - zlffz sin 0 cos# = 0, (7.2-13)

with the boundary condition

= û .dr .+.oo. I(7-2.14)

It is easy to integrate such an equation once, and obtnln what is lœown

14O BASIC MICROMAGNETICS

as a Svsi ïntvml lt can be aeieved for example by mitipyng the dife-renllxl equation by d8/dz aad iategzating oMc z. The r-lt is

2l ds a 4-C - - Aez sin 8 - Ka sin $ = const,2 d.'r (7.2.15)

where the right hand side is an intagration constant. This integration isactuvy a particular case of a general theorem (2701, whie,h will be provedin section 10.2, according to whicà all one-dimensional problems in staticmicromagnedcs have at lMst one 6mt integral.

The integration constant can be determined lom the condition that thestructure must be a OCJJ separating two domnlmK maaetized aloag Lz. Itimplies that sin # = 0 at z = +txh, and when this condidon is suuitutedin eqn (7.2.15), together with eqn (7.2.14), tMs conennt is s-n to be 0.Hence

dê :?./ ftru z- = + 1 -h sin $ sin 0.gzt T' c T zk% (7-2-16)lntegratîon of thss equation is obvious for either choîœ of the si> in frontof tlle square rootv and one of the branches is

1 + n tzmh @/J)cos $ = . - = mz,'F 1 + mtaah2 (z/J)

J = j/' ,2A%Kz

N = - ..K'l

(7.2.17)Actually, insteo of z the argum%t should contaân tr - z:, where zo ks theseond integradon constaat of the original second-order diferential equa-tion. However, the origia does not have any men.nipg in an ïXJP,A crystal,and zo may be omitted.

The wall erldr.o rztn also be calculated analytirolly, by substituting eqn(7.2-17) in eqn (7.2.1$, and carr.ying oqt the integration. The wall energyper unit wv a.Z'I?A is thus found to be

Fr +

-1 + n c--antz-)s ) , (7.z18)s = azfzc arW .L

whic,k depends only on t'he Aninokopy and Gchn.nge constants of the mate-riak The spontaneous magnetîzation, Ms, does not enter, because it is onlyconnected with the magnetostatic erlerr term which ltM 1:ee.11 eliminated1om tke present calculafon.

Thœreueldly, the magnetization in eqn (7.2.17) becomes parallel to+z only at infnity, and the wall IUA aa in6nite width, but of course thisinînity need 'not be taken too seriously. The scale of z in this equation

'

is 6 = Cj2Kït at lemst when x is smal), and thln expression is usuallydefned ms the mall WdfY Any remsonable denstion of the width as the

MAGNETOSTATIC ENERGY l41

distance ove.r which most of the rotation âom mz = -1 to mx = +1 ukesplve, wttl lead to something of the order of this quaatity- A more accuratedeNnition is s'ivea in (27$.

If the anisotropy is cubic, as is t:e case, for raxmple, in iron or nickel,there are three easy axes along the three cubic axu. The mMnetizationin some of the domains is at 900 and in some at 180* to the one in theneighbouring domm-n (sœ Fig. 4.1). The structure and enera of both the90* an.d the 180* wxllq have been calculated (r9) in a similar wa,y to thecakulation in this section, at lemst for a negli#ble Xa. The results are alsom-mllar to the foregoing. etœtriction, which has been added (2791 asa uninNn-nl Anisotropy superimposed on the cubic one, has some efect onthe wall structure, but its efect on its enera is negli#ble.

The calculation of the energies of the dferent walls is the lwq-qis of whatbecpme known E145) as the domain tàeory. ln calculathg the eaerar ofdxezent confgurations of domains, the walls llet-wsvm the.m are takn tohave a zero width, as in the calculation of the magnetostatic enera ofthe two domnsns in section 6.2. But then the enea'a of tphe walls is added,using upressions such as the one for the unlim-al aaisotropy in eqn (7.2.18)here, and multiplying by the wall area according to the assumed geometry.This tecxnique azows the comparison of the total eneror of a11 soz'ts ofconfgurations, in an attempt to fnd (5% 279) the one whose energy k lowerthan that of the others. It is even possible to add the interaction of eachconfguration with a.a applied magnetic feld and try to follow theoretically1he whole hysteresks c'urve. For large and complu systems it is the onlytheory, and these stùdies still continue (28% 2811 today- For small pardclothere are better and more reDable .methods, whie.h will be decribed iniapter 9, but this technique is being used (282, 283) for them ms well. Moreabout walls will be given in chapter % but mœt delzu-lK of the domm'n theozyare outside the scope of this book. Only before conduding this section, thereader must be waamed not to be msqled by the elegaace of the solutioninto believing that the calculation presentM here îs the fnal result for thewall structure or its energy'. Even a Iarge crystal ends somemheat and thestructnre presented here create much too much charge on the surface.Thks cxarge tdemaretizes' the wall and distozts its s/ape to reduce itsma>eœtadc enera, and this distortion propagat% into the internal partsof the wall. The resulting structure becomes quite complex, and eztnnot beexpressed by a one-dl'mensional ftmction of spaze. Thewhole problem thembecomes much more complicated than the one presented here, but thencomplication is inevivble in ferromaretism.

7.3 Magnetostatic EnerrThe magnetostatic enerr term has b'een inkoduced in section 6.1, ms eqn(6.1.7), but it lm.q not been prove there. lt will be put here on a sounderbmsis tlun the ent Zven there. In order to satisfy those who may

142

fee.l uneasy about a mere acceptance of Maxwelps equations ms they are, itis necessary to start from the atomic nature of real 'mxtedals, which wasnot even Hown at the time of M>xwell. The approvlmation of a coneu-ous material is inevitable at the end, but it is importaat to nodce that itis not inednced as an arbitrary mssumption. It comes as a well-justifedapprolmation for the lirnt't of vadual variauon over a <ze which is largecompared with the lattice constant of the matem-nl. lt is n-xrly the same

justifcation and the same limit as in the classical appromadon for thee-xchxnge enera in section 7.1. If anythlngj the approzmation for the mag-netostatic term is ev> more justifed than that for the exchange tRrm , aswill be seen izl the following.

BASIC OCROMAGMTICS

7.3.1 Physially Smcz! SplttrtConsider a lattice made of magnedc clipole, with the magnetic mom' entth at the lattice pnint ï. Let h.. be tàe feld intensity at tàe lattice pointï due to all the otlter dipolœ. Ia the absence of thermal fuctuations, thepoteatial enera of thiq is

1Cu = -j' Y! 1% - Y', (U-3-19)

1

where the factor 1 is introdux because the surnmation contains ea.c.1z of2the inteeractiolks tWce: once as the interaction of the dipole 6 with the felddue to .h and onœ as that of the dipole .i wtth tke feld due to ï.

Let a sphere be drawn around the lattice point f- If its z'adius R is largecompared wit,h the unit cell of the maœial, all the dipoles outside thisspheze may be taken as a continuum for Yculating the feld whiclz theyœeate at this pardcalar point, ï. Therefore, the feld lu at this point maybe eviuatM by tnlclng the ield due to a continuous matarial everywhere,subtntcting from it the feld due to a continuous material inside this sphere,and addiqg the feld due to the dîscete dipoles witMn the same sphere.The irst of these terms is the feld calculated in section 6.1 1om Mmxwell'sequadons. It will be denoted from now on by H', in order to keep thenovtion H for the applied âeld due to curren? ia some exteaal coils. Azhas already been explained izl sedion 6.1, these two felds may be taken assepara* ent-idœ and thea superimmsed. It is necessary to subtract fropthis feld H' the contribution of a continuous mMnetization inside th1sphere. If this magnetizadon does not varqg very mucà inside the sphere,the latter Neld is approzmately the demagnetizing feld of a homogenmuslymagnetized sphez'e, given by etw (6.1.1.3), namely -(4?/3)M. Hencl

42: ' -

hs' = H' + M + ht. , (7.3.20)7-where h? is the conkibution of the dipolu inside the sphere.

MAGNETOSTATIC ENERGY 143

'àhe use of the demagnetizing feld is jalh-ia-êed if 'tàe radius R of thesphere is smaz compared with the scale over whicà the direction of themagneœation can be tnlrnn as a const=t, or at most as a linaar llnctionof spaœ-lt is necessav to mnw sare that this assumption is not in conlctwit,h the frst asstxmption, that R is much lrger thaa the lattice constantof the material. The second requkement is that R is small compared withthe sœenlled ezthange Jenvl/j, wàic.k is the length over which M cllaagœ,namely sometMng of the orde.r cf the Lzandau and Lifshitz wall width,

Cf2Kï. For a typical case of permalloy, with C x 2 x 10-6 erg/mzz audKk ;4$ 104 ezg/cap, this wazwidth is ahmt l00am, amely about 300 unitcells, and about the same number applies to iron. ln these m-rt- it is indeedpossible to defne an intermedln.te value of R, 611611img both rmuirementfor being suEdently hrge and mzRdently small, which is n.sawnlly referredto as a pltysicazy Jzncll sphere. It should be e'mphasieM agna'n that thispossibûiv of de ' such a physienlly small sphere is due to the e-xchaagelwqng very strong over a short zange, keepîng the spins almost aIig'IIC'd overdistaacœ of the order of a unit (e- There are casœ of certain rare earths,or thdr alloys, for whic,h Kï is muc,h larger and the exchange length isonly a few lattice conKunts. In thœe cmses the contîauum approach is aotjustiâed, and it is aeceasazy (284) to coasider a fnite Gaage in the directionof M 1om one lattice point to the nexk .

The last term in eqn (7.3.20) is a sum over felds due to dipoles,

Jz. 3(../ - riylri.f( = -..-.J- + , (c.:.).a:.)hz a sIrïjl IrzgljTéi 2<2

where z'Li is the vector pottin.g 1om lattke miat f to lattice point j. ln aphysically Kvnxll spherey gj is Rtually a constant, which does not dependon #. In thks rzuej it is pvible to write, for examplej the z-component ofthe feld in Cartelm coordinatœ as

g. 3ze(#..a4y + lAvyij + #ùx.r.f.ïlà,' = - + .G za r.sLj ij(7.3.22)

U the crystal àas a cubic symmetry, a snrn over a sph'ere of the term withajtwj or with zijzzi vaaishœ because there is an equal contribution 1omthe positive and negative term, =d. actually this statement is true foralmost aay other symmetry. àlso, for a cubic s'mmetrs tr, y and z areinterchaageable, and therefore

.s'?, & ? z? 1 z ? + y ? + z ?é 1 .-w :y'y v.aj. = y..y zi uaz y.y ,J m .y'y .y.. lj . ..y g w. rqr.g.zz;r'F rq rq 3 P 3 %<J xj eJ IJ

Thns the total snm in eqn (7.3.22) is zero, and so is aay other compoaeat of

144 BASIC MICROMAGNBTIV

eqn (7.3.21). In a nonmbic smmetry the sum is not zero, but it is obdousfrom the form of eqn (7.3.22) that pz ca.n be taken in front of a sum wllic.kis just a numbec, and the same is kue for the other componeztts. On thewhole, under the anmption tkat M may be approzmated by a constantinside the physically srnzzl spkereo h; is a liaear function of the componentsof this M, with coeRcients which dep%d only on the crye-xll'oe symmetry.ln other words,

14 = A . M , (7.3.24)where A Ls a tensor which depends on the czystalline symmetry and whichvanishes for a cubic smmetzy. Substitlzeg cxqas (7.3.24) and (7.3.20) ineqn (7.3.1$, and chan#ng the sam to a.n integzal, the magnetoMatic energy

1 t 4/Su = -g M . H + SM + A - M dn (7.3.25)where tke intwation is ove.r the ferromagnetic body.

It must be aphmsized that tke approfmation of a physically smallsphere does not really require that M is a constant inside 'iltiq sphere- AJfnecr c'hxnge over the dlrnensions of the sphere will not makeany dxerenœto the foregoing, bfvm.mse it is easy to see by smmetry considerations thatits contzibntion is zero. A genernllmation (285) of the foregoing derivationconsidered the case of a quadratic change over tke sphere, and showed thatîts contribution is also zero for a suEciently Mgh czystalline symmetry. Fora lower smhetz'y, the contzibution of a quadratic term is not zero, blzt RIUA been slloW (2851 to be negYbly small for all cases of interest. Theproof of this theorem can be summarized qualltatively by the followiigvgument: if the change of the magnetization is rather slow, the foregoingks cozrect. If it is not slow, the maaetceatic energy may be assnmed tobe small compared witk the achange energy, and a certain mistake in thesmaller term does not aeec.t the total enerr. The physically small spheremay also be genernl-lqed (285) to be a,n ellipsoid, but thks gencxliqation doesnot have any reeal eeect on the proent calemlxtlon.

The middle term in eqn (7.3.25) contains M . M, which is the constantM'zs that depends only on the temperature, and does not depend on thespatëal distribution of M. Therefore, it is omitted, whicx only means re-decning the zero of the magnetostatk energy and has no H'H on energ.gmsnlmiqations. The lmst term Ls a.n energy denstt M . A . M which %Mtke s=e formal form of tke anisotropy enerr density discqssed izz section5.1. Therefore, it may be included in the anixtropy energy instead of here.lt is particularly convenient to .do so because the anksotropy constank inm-t ca- are taken from the eerimental values, which already inclvdethis term. It should only be noted that when the sph-orbit interaction iscalculated fl'om bmsic principles, ihis term should be added to the nanltingesotropy. The magnetostatic energy has thus been skown to be

MAGNETOSTATIC ENERGY

s. = -) / M . s' d.,

l45

(7.3.26)

whic.k is the same =pression already used ir chapter 6 without proof.M was the cmse with the exckange energs the physirmlly DaII sphere

ks assumed here to be eatirely inside the ferromMnetic body, and this as-sumption fils for lattice poiats near the surfnce-Here,p' az't of the necessarycorrection ks already iacluded in the surfnne cltvge, Tven in section 6.3 aspart of tEe chssical enera calculatlon. The resi of this error Gects onlyspins wkic.h are qlzite close to the surfRe, ard can be expressed (145) Va term <th the same funcuonal fnnn as the surface aaisotropy eneraYrm. Therefore it rAn be txl- into account ms anotker contribution fo thesurfve aisotropy.

The magnetostatic energy a,s exwessed by mn (7.3.26) ks non-locatThe volume integral in this equation contains H', wbich in turn ilM to becvaluated by aaother volume inteval (see secdon 6.3). lt efec:vely meaasintegrathg twice ove,r the same voblme. In tzhis respect this enerar term isvery diferent from the excbaage aad aaisotropy energy terms, which arelocal, aamely involvirg oae volume-intevatioa of an enera density. Thisproperty is another a,ped of the long-raage nature of the mMnetostadcforces which requiretaldng irto account the inlradion of e-qnh dipole wlthevery other dipole in the ferromoet. There Eave been some attemptsto approfmate this double intevatîou by a slngle one, and these attemptshave failed, as revicwed in (270J. The poiat is that, in prhciple) a long-rangeforce cannot be replaced by a short-raqge one wlthout loshg some of it,simportant propeies. Therefore, the Omplication of a six-fold infagrationis paz't of the physical problem, aad ms suc.h it is hevitable.

7.3-2 Pote A'/zoïdcnœ Pdneïple'I'hetx are other forms to ecpress the magaetostatic enerr, which are math-ematically equieent, but one may be more usehl tkan the otber ia someproblems. In order to establisk them, it ks necessary frst to prove a theoremconcerning the magnetic inductioa,

F = H' + 47rM, (7.3.27)whiclt has already ben defned in eqn (1.1.2). lt is denoted here ms B? inorder to emphasize that it is only the part of B whicà is related to H?, aadnot to the applied feld H.

According fo eqa (6.1.2))

J H? . B'dr = - J B' . Mudr = - j (V . (t7B') - U% .B') dr, (7.3.28)

where the secoad equality is an identity, aad the intevation is Msumedto be over a large enough volume to contain all the ferromagnetic bodies.

146

The second tmrrn in the last exprœsion w.ninhe vcording to eqn (6.1-3),and the 5.rs-t term rxn be traasformed according to the divergence theorem.Hence,

H! . B'dm = - n . UB/dS, (7.3.29)wheze n is the normal. Now, the boundo conditions of Maxwell's equardons aure a contiauit.y of both U aad B= evezywhere. Therefore, thesurface integrals cance) when evaluaM on b0th sides of aay surface of aferromagnedc body, and the rïgbt-hand side of eqn (7.3.29) is au iategralover tke outside surface of the volume which hms Yn assumed to containa2 these bodies. If tMs snrfxce is allowed to tend to infnity, outside allferromagnets B# = H# = -VU, which tends to zero at least as fmst a: l/r2(see the boundazy coaditions in section 6.1). Therefore, UBs tends to zeroat lf'xqt as 1/0, wMle ds incremses as rZ, and the whole integral on theright-hand side of eqn (7.3.29) tends to 0 at înAnity, wMc,h mpltns that

H' . B'd'r = 0.a1l spaco

(7.3.30)

BASIC MICROMAGNETICS

This theorem is of some interet in its own right. But it is x.lKn n-fulfor a transformation of the expression Tor tlle maenetxtatic energp ït isae-etine by substituting ma (7.3.27) in eqn (7.3.30), whic.h yields

H! . (H' + 4rM)d'r = 0.all syace

(7.3.31)

Brexlclng this ixtegral iato a mym oî two integrals, it is seen that the onewitic.h 'contains M is proportional to the integral for the magaetostaticenea'r in eqn (7.3.26). To begin with, the htegral in eqn (7-3.3:) Ls overa.!l space, which hclude Nrts in which there axe no ferromagnetic bodiu.However, M = O in those parts of space, and the.r do not contribute to theseond part of the integral. Therefore, this second integral is also over theferromagnetic bodies) as is the integral ia eqn (7.3.26). Qemrrrm#ng,

1 zaSu = - S dr-81 ul spacc

(7-3.32)

This form of writing the magnetostatic eaerR demonstrates the polzcfoïdcntz zdnesplc. The intisgraad is positive evezywhere, which makes themagnetostatic enerar azways positive. The smallest possible value for tMsenera term is zero, azd this <ue rztn only be weieved when H' is lden- ...

tîcazy zero everywhee. Therefore, the maaetostatic enera term alwaystries to avoid aay sort of volume or snrfazv charge. A complete avoid=ceks not 'lzsually pœsible, unless the geometry is that of a toroid, in which

MAGNETOSTATIC ENERGY 147

a magnetization with no divergence ca.n be parallel to the surface- How-ever, the prindple is that this enera term tries to achieve confgurationswith as little chazge as possible. This prindple has already been used inthe qualitative arguments 11/ section 6.3, which showed, for example, thatan ellipsoid would rather be magnetized along its longest afs, etc. How-ever, in that section this argument was a little premature, because a cleve.rreader may have wondered why avoid the charge rather thaa think of somesophisticated arrangementé with a combination of a positive and negativecharge, whose enerr may be lower than that of no charge at a11. Only now,afler the proof of eqn (7.3.32), with an integraad which is always positive)it should be clear that such a fsophisticated' arrangement caanot efst.

There is still another form to express the magnetostatic energy termjwhich is also derived from eqn (7.3.30), which can be written as

'

(B' - 4xM) . B'dv = 0,+1 space

(7.3.33)

in accordance with eqn (7.3.2/0. Rearrangingz and using eqn (7.3.27) again,

k 1 c 1 1- B' dr = M . B'd.r = M - (H' + 4'mM)d'r, (7.3.34)B'I au spaco i' i'

o r

where the intevations on the right hand side are (wer the volumes in whichM # 0. Substituting from eqn (7.3.26),

1 ,a a- B dr = -1'M + 2* .&f dr.8* wll spaco(7.3-3 5)

And since /./2 is the constant Afos ,

(7.3.36)

where 'Z is the volume of the ferromagnetic body, or bodies.It must be emphasized again that an energy calculatedfrom eqn (7.3.36)

for any pazticular case is going to yield exactly the same numerical valueas the energy calculated from eqn (7.3.32), because these two equationsare mathematically identical. However, the msnus sign in eqn (7.3.36) doesnot allow any pltysical interpretation of what confgurations of the magne-tization this eaerr term prefers, which ca.n even come close to the simplepicture of pole avoidauce implied by eqn (7.3.32). lt can be said that the

,2magnetostatic enera term prefers the average B to be as large as pos-sible, but this statement does not help at all to see the actual preferabledistribution of the feld B;, or that of h1:. lt has been claimed (286) that

148

eqn (7.3.36) hms to be prefen'ed over eqn (7.3.32), because B' hms a moredired physical meaning than H/, which is essendally the same ms sayingthat surface or volume charge should =ot be used because there is no phys-ical meaning to this charge. Sometimes a pure mathematical concept eAn

be more convenient, and allow a better physix intuition into the problem,than a true physiY approMh. The same disadvantage of a lMk of physitzalintuition applies Jdlrsn to other forms (287q of the magnetostatic energy term.

Beginaers may wonde why the frst term i.a eqn (7.3.36), which is justa constant, is not omitted by redef Tn-ng the enerr zmo, ms hms alreadybeen done several times in this book. Of course, it is quite le#umate todo sq as long ms tYs new defnition is used tonsltently. However? it is notdone because it is not usefnl, ald will only mislead people to bezeve thatthe mavetostatic enerr prefers B' to be as lazge as possible evenrwhere.Redl6m-ng the zero wûl not change the mathemativ fact tàat wbateverB' is, the newly defned energy cannot pcebly be more negadve than-2rcMs2ïzr, which is the enerr of a conouration with no volume or sur-face charge in the new system. Defniuons are chosen to be helpfuly andconfasing deim-dons are better avoided, even îf they are quite l4gitimatein prindple.

BASIC MICROMGNETICS

7.3.3 - rociiyA vezy mwerfal tool for calculating the maaetœtatic enegy of œrtnx'nconfgurations can be obtained Fom a gen-mrwzkdon of eqn (7.3.30).

Consider two distributions of mMneœation in spaœ, Mz and Ma. LetH) be the maoétic feld prodqced by Mç, for ï = 1,2 respectively, and 1etB) = Ht + 4xMç. Uing the sxrne proof used to prove eqn (7.3.30)9 it isreadily en that

H'z . B'adr = H?a . B'zdm = 0. (7.3.37):111 space rzll space

TEe properties of the functions msed for proving eqn (7.3.30) were that E'Ls a gruient of a potential whiclt is continuous everywhere and is regularat infnity, and that B' is continuous everywhere, ar.d all these propetiesG .

are aâso fulflled by H( aad Bt separally. Therefore, the proof is the Knme,and ia the same way ms writin.g eqn (7.3.31) it ks pfwNlble to conclade thatboth l

H'z - (H1 + 4'rM2)g'r = 0 (7.3.38)a11 epace

andRc' . (H1 + 4rM1)d'r = 0. (72.39)

a11 splce ?

Subtracting 'th- two muations, tàe part with Hl . 1% is common to b0thetpre%ions and caacels, leaving '

MAGNETOSTATIC ENERGY 149

(7.3.40)

' Of course, it is never actually necessary to integrate over the whole space,and e-ach integral is over the volume in which its integrand is not zero.

This equation is known ms the r#llrodty theorem, and is very use-ful in solving magnetostatic problems. The integrals in eqn (7.3.40) areparts of the integral in eqn (7.3.26), if M1 and Ma are parts of the totalmagndization distribution, M, and the theorem is true for auy arbitrarysubdivision of the magnetization into these two entitiu. Thereforep an ap-propziate choice of the way in whic,h M is subdivided into Mz and Mamay often simplt the evaluation of the magnetostatic eneza. Of course,this choice has to be ftted to the particular case under study: and thereare no guideznes to faetate the decision. An example of this use of thereciprocity theorem will be given in the derivation of the Brown diferentialequations in section 8.3.

Jn principle, the redprocsty theozem is not limited to Mz and Ma whichadd up to the total magnethation distribution, M. The proof of this theo-rem is quite generalo and applies also to cases in which Ml and Ma overlapin some part of the space. Eowever, nobody has ever used this theorem forsuch an overlap. The most direct application of the theorem is for the casein which M1 aad Ma are the magnetizations in t'wo separate bodies, andeqn (7.3.40) is interpreted to mean that the interaction of the magnetiza-tion izt one body with the feld cre-ated by that of the other body is thesitrne as the interaction of the magnetization of the second body with thefeld cre-ated by the drst one. The most common evnm pIe (288) Ls in thecalculation of the energy of interaction between a recording head and thebits it records on a disc or tape, or in calculating the signal on the readingheadj which învolves the same integral as in eqn (7.3.40). It is rathez easy toknow the feld due to the magnetization in the head, and the magnetizationdistribution in the recorded fape. lt is much more diëcult to estimate theîeld due to the recorded tape aad the magnetization distribution in thehead. Equation (7.3.40) makes it possible to How the interaction withoutevaluating the more 35fB cult part.

Brown (1q has listM this reciprocity theorem as only one of severaltheorems to which h4 gave the general name of reciprocity theorems. Mthe others are less common, and less applied, 5n the literature, and areoutside the scope of this book.

7.3.4 Vyyer and Lover SdtlndsWhen it is diEcult to evaluate the magnetostatic energy exactly, it mayoften be suëcient to have a reliable estimate for its value. An appremationmay do, but a'a approvimation is reliable only with a good estimate of theerror involved. J.n principle, the best estimate is obtained when the exact

150

value can be put between two bounds, esperzxllywhen these two bounds donot dlfer very much from each other. Somekime suc,h bounds are found bya certltin t:1c,1: wlzich ks applicable only to a particular problem. However,for mavetostatic enegy calculations, 'Brown (2891 has devised a rathergeneral method for fnding both an upper bound and a lower bound. J.fproperly used, these bounds may be suhdently close together, so that theexact value may noi be needed. Thee bounds will be swcfed here, butthe proos that they are indeed a lower and an upper bound is diferent1om the one orisnally pr-ntv by Brown. The latter was not very easyto follow or to unders/-xnd.

Let M be the actual dkstdbution for whick the magnetostatic energyis to l)e calculated, and let H? be the true feld due to t%u magnetization.Let Hl = -V . t: be the feld due to some othev distribution, whic.N will bespecïed later. Obviously,

BASIC OCROMAGMTICS'

l , / , zEv >- Es = Sxt - - k H: .. - H ) dr ,8x .u xpa.r(7.3.41)

where the inequatity results fzom subtraztiag an inteval whicb cannotbe ncgative, because of the squaze. Opeming the brzmlrets in the in*gran;using Grst eqn (7.3.32) then eqn (7.3.27), aad omitting the 'all space' whiehis implied from now on for all the integrals in this section,

su - sl (2Hk .n' - Hl2) d.r - sl g2y,q . (B' - 4,mM) - Hkj dv.

(7-3.42)The pazt of the integral which contna-ns Hk . B' is zero according to eqn(7.3.37). Note that the proof of that equation rmuized oaly that H) isa gradient of a potzmtal which is conthuous and regular at inflni ty. Itis not even necessazy that this poteatial is due to any real magnetizationdistribudon. Therefore, by writiag this potential eelidtly and substitutingeqn (7.3.42) i.n eqn (7.3.41)

su z su - /M .vlaz- sl /(v+)2 dp, (7.3.43)

where the fzrs't ineih ove,r the fromagnetic body (or bodias), and thesecond integral îs over the whole space.

This retzlt provides a lower bound to the correct maaetostatic enerrCv of a given magnetization distribution M, in terms of an arbi-vy func-tîon of spaze: +. The only limitation on tke arbitrar.g Goice of + is thatit is coneuous everywherw aad that it is regular at inCnliy, because onlythœe properties were used in the proof of eqn (7.3.43)- A discontinuity oftàe dezivative is allowed, aad may be introduced anywîere- However, itis not usually suEcient to have just a lower bound, which may be cozrect

MAGNETOSTATIC BNERGY

but not useful. After all, a zero is also a lower bound to the magnetostaticenerr, which is always positive according to eqn (7.3.32), but this lowerbound does not help to solve many prèblems. A useful lower bound ts onefor which Cn is not very dxerent from Cu, which is intuitively understoodto be more lgely when 41, is chosen to have at least some of the featuresexpected 1om the real potential of the problem, U. It should be noted thatif 41, = U, the inequality in eqn (7.3.43) becomes atl equuty, according toeqns (7.3.26) and (7.3.3$. Therefore, the best choice should always be a41, which approam 'matesj or at least imitates, U. Thus, for evnmple, it some-how does not seem right to choose a function + which has a discontinuousderivative inside the ferromagnetic body, even if such a choice is allowed inprindple, am.d even though it has never been proved to be a wrong choice.At any rate, such a choice has never been tried in any of the applicationsof this theorem iu the literature, ms cited ia (288j.

In practical applications, Cu is not known, and it is impossible to de-termine how good the choice of 41, is by checking whether sz is close toSu. Thereforej a lower bound by itself does not help at all, and the onlycriterion for the useftllress of 4 is when an upper bound can also be foundthat is not very diFerent fzom Sz- Only in such a case e--in one claim thatthe exact energy value Cu is suhciently well deternzined, because it must

!) .'

be between these two values. The importance of Brown s bounds are thusin the combinadon of 30th of them, and not in each of them by iiself.

To obtaln an upper bound, a positive integral is added to the trueenergy, Su ) in the form

1 , 2Cu f f's = Cu + (B1 - B ) drt (7.3.44)8*

where Bz is a,n arbitraryvectorialfunction of space. It can be seen 1om theproof of the relations used in this derivation that it is suhdent to requirethat B: is contiauous everywhere, and that V . Bz = 0. As is the case with.ï', this B1 does not have to be connected with the real B' of the problem,but it helps if they are not too diferent. Substituting fl'om eqn (7.3.36),opening the brackets, and ushg eqn (7.3.211,

1 2 zl's = 2.mM.27 + gB1 - 2B1 . (R + 4rMj dr. (7.3.45)8x

According to eqn (7-3.37), for which a11 it takes to assume (as mentionedabove) is that Bz is continuous and that its divergence is zezq

1 cI'M < En = 2rMs2y + By dr - B1 .M dr, (7.3.46)8,r

where the last integral is ove,r the ferromagnetic body, and the one beforeit is over the whole space. As before, V' is the volume of the ferromagnetic

152

body (or bodie). And, aq in the case of the lower bound in eqn (7.3.43), theixequality becomu an equality if B: = B' due to the actual magnetizationdistributioa M.

The constraint V . B1 = O is enlrlly implemented by càoosing Bz =V x A, in which cxe the vedor poteatial A is almost compldely arbitrambecause it is vezy easy to take care of the requircment of continuity in anyanalyts.c model. lt is always preferred to defne some adjustable paraïetersi.n both the scalarpotendal 'I' aad the vector poteatial A, and mnvlmlee àkaad minimîze SB with respect to these parameters. This l-Anique ensuretkat the best upper and lowe: bounds are obtainedfor any cNoxn functionalform of these functions of spMe. With some htuition, or some luck, theupper aad lower botmds may be clckse cnough to make it unnecessaty to gointo the Ompatations of the actual magnetostatic enerr, and some suchcmses have been reported (288J. A dferent cmse wi.ll be given ln section10.5.1.

H this section, integrals over the ferromagnetic body (or bodie) audttegrals over the whole space were trp-qtel on an equal bGs, even usingthesamesymbol for both- ln pmdice there is averybig diference betweenthae two integals when it comes tzl numerical computations. Because ofthe long-raage nature of the magnetostatk potentials, the integration out-side the ferromagnetic body converges very slowly, and it is necessm to usemany times the volume of the ferromagnet before the result can apprH-mate an integration to in6nity. It must always l)e borae in mind that whentwo expresbions aze identical mathEmatirnlly: e.g. as aze eqns (7.3-26) aad(7-3.32), their computation shodd only converge epenttt4ll: to the >menlzmerical rault for the same problem. lt does ztt?t mean that they take tkeRxme time to compute the rx-lae result. BeAuse of the slow corvergence,sach a numerical integration outside the ferromagnet haq nevc been con-sidered pzactical 5n a=y of tke applicadons of this theore,m reporfe PM!so faz, with only one exception whiG will be discussed in section 11.3.4.Tncfmz!, the potential was always taken ms a ftmctionat fo= for which atleast tàe contribution to J(V+)2dr oz J B2zd'r from the m,rt outside tkeferromagnet could be carried out aqazytkal'ly. More details rltn be foundin the references cited in (2881, and it can only be Mded that there is acertain suggestion (290) for a-rather general nlnKs of Nnctions whic.h eztn

be used for this purpose, b0th for the scalar potential * aad for the vectorpotential A.

BASIC MICROMAGNETICS

7.3.5 Planar J?ectangleIt hms akeady been mentioned in section 6.3 that the formal solution of eqn(6.3.48) involves integrations which rAn be very rarely carried out xnnlyti-cuy. The main reason is that the nllrnerator is a Tnndion of r' only, whilethe denominator involves r - rJ. These expressions are rlsmcult to mix, evenfor rather simple functions.

MAGNETOSTATX BNBRGY 153

Consider i.:z particular the cmse of a ferromagnetic body in the form ofa prism, -'c % z i % -ù K y K b and -e % z K c. The poteatial dte tothe volume charge; namely the îrst tntegral in eqn (6.3.V), can be wdttenin Cartesian coordinatu ms

t'!'h.?s fa'' '4' a') am (x? .p? z') 4%a. x? %l' z/a 5 q. * x 1 . 1 M *' *' œ q .+ - + ,r ::7r' @vz zLvoluma = '-JWS - -

..c -.ô -. (z - z')2 + (# - y')2 + (z - ztjzz z J.

'

x dz dy dz , (72.47)where m is defned in eqp (5.1.6). Intevatiag by partsj the frst te= withomzlox' with respect to XL the seœnd term with respect to &?, etcu it isseen tàat all the Gmrusions bdween tke lt'mits -4, and d,, etcp cancel theappropriate terms of the potential of the surface ckarge, namely the secozdiutegral in eqa (6.3.48). For tàe reGer who has never t'ied this b'nd ofexerdsel I highly recommend following the details of the laat statement,wàick is the bet way to understand the meaning of the normal n. At anyrate, the result is tàat the total potctial due to both suface and volumecEarge is

c t) e,

&r(z, y, z) = el/k..c -: -a

(z - ,'zJ)mx(z/, y', z') + Ly - ytjnhilz'tfnz'j + (z - z')ma(z/, /, F)x --

' 2 + ( - #')2 + (z - z')2)2/2E@ - tr ) #

x dW d/ dz'. (7.3-48)If m is made out of (rather small) htegral powers of z', f and z', all theintegrations in this equation can be cnm-ed out analytically. For prnztta'callyàny other fundion of these vuables it ks impœdble to do anytldng analyticunless the mixture in the denominator can Kmehow be tm.nqformed into aproduct of functions of x and z', and so fortà.

No general trazsformation of this sort ks known for the general, three-djmensional integral in eqn (7.3.48). Howeve, a general twmAormationis known for the two-dimensional rmse, whem m does not depend cn z,which can be either becausd the snmple Ls a vezy thin fhn , with c -/ 0,or because the sample is very long in one dîmension, and c --+ cx). H thetwo-dimendonal caseit is known from any undergraduatetextbook that thepotentiiof a unit charge is 1og(Va)2, instead of the three-dimensional 1/ruscd in tàe derivation ()f eqn (6-3.48). Repeating the foregoing întegratîonby parts for the rectangle -4, K z K c, -5 K y K b leads to

154

6 * @ - z')ru(z',&') + (v - y'tmvlz', :/) , ,U(m, &) = 2Ms , a ,;c ' dm ## -

-i -. (z - z ) + Ly - v .

(7.3.49)H this clme tt is possible to use the wemuown Laplce trxnxformj

* o g ja .g/ jcœ((p - v'ltle-lm-m 1 * = - (7.3.50)@ - F)2 + (y - F)20

BASIC MCROMAGNBTIG

(and sîmllxrly for t:e second term), aad zewrite eqn (7.3.49) inside ther--xngte i'a the form

G c, y: ; -(Z...m).& yytg tj l +. pmjtz - z/ljj mvtz) ylj''- 'CN'2(3 AV )d # ,

-(v-v#)z g l (zl Jy-(T/ -v)z g t gzz Jj (g aogj.;X e # - 'NV , # # ' *

y . '

TH expression was ori/nally derived (291) as theliml't k -> 0 of a periodicz-depeadence of the form costkz), directly fzom the A:11 three-dimensionalpotential in mn (7.3.V). It should be noted that brpztMng the integrals ft)ra? > z and z' < z taald m'mllxrly for yt) is due to the absolute 'value in mn(7.3.50), a) that the result presented ia mn (7.3.51) is valid only for thepoteadal indde the rectangle. If dther z or ?/ Ls outside the ferromagneticrectangle, tkis breakng down into two integrals has to be modifed, andHiFerent exwessioas have to be îtted according to the quadr=t oqtddetke rectangle for which the potentsal is calnnln+M . These distiactions arenot usually necœrars because the potential in the rectxangle, -c S tz % aand -5 S y S b, is snmdent for calculating the magnetostatîc enera.

Repladng a double integal by a triple onemay not sea a good stat a frzt glance. However, the advanvge of eqn (7.3.51) ()#er eqn (7.3.49)ks that the former contains trigonometzic and exponential fanctions whichare readily upressed ms prodncts, namely a, function of zl times a ftmctionof z, and similrly for y and 6 In a product one is highly likely to be ableto perform botk întegrations analydcally, for a wide variety of hncdonsm, which cannot be haadled in the form which coatains z - z? in eqn(7.3.49). For the maaetostatic enerr in two dimensions there is a foar-fold integration, which is transf ormed here kzto a fve-foid one. But if fourout of the fve caa be perfozmed analytically, the nnmeric.al integration of .

the remiaing 11z*g1-a2 over t is much simpler than a four-fold numcicalineration. rnis ttenique has iadeed beeu found useftzl in the calculationoî several casœ, dted in (288) .

MAGNETOSTATIC ENERGY l55

The subsdtution in the enerr will only be demonstraM herû for theparticular case of a maRetization which does not depend on y. Ia this cae,after carrying out thû intûçation over yl in eqn (7.3.51),

x :-(:-v): - :-(y+v)t a

Dk@, #) = Mfs xs((z - z')t)mv@') &'1 -.0

cosLyt) siatsfl 2 ? -(z-xz)t dzt+ 2 'cutz let -.

, -(xr-m)v aqy cg- mc (z )e .

r(7.3.52)

According to eqms (7.3.26) and (6.1.2)j tite magnetostatic enerc of such aone-dHensional magnetization structure in a rectangle is

1 1 * b t3rJl= 8USM = j. M . TU dS = -Ms a.@) + mv@) m dz dw2 -. -, dz %(7.3-53)

Substituting fzom eqn (7.3.52), aad - g out the integrations over y,

Jœ' 2 2 2 , zDu = 2j%f. -z sia (btj - Emxtzlq - mstrclt.3 - u.

tz :. .n-zbt cr - I z. -. a ' I z gm , a,g + rzz t z %x 'nzz (z' )e s. )t .-.o-@

j' mv(z') c.ltz - z/l4 az'tfzj a.X (7.3.54)

This eoression can be simplifed by noting that.= cosltz - z')à) .-zyf ; (jj .z

* sinztôt) -jzwxyjt a (ya.55), jl - e j e ;t c0

because each of thee ecpressions can be found in tables of inteval trans-forms as being equal to

1 452log 1 + .i' @ - z')2

lt is nöt adeable to use the lattûr expresion bdore iategrating ove,rz and z' for specïc magnetization covgurations. Howcxver, eqn (7.3.55)allows combining the two expressions together. By subsdtuting it and theknown intagral

156 BASIC MICROMAGNETICS

Q= sia2@) gb1 t 2 df = 'i- (7.3.56)in eqa (7.3.54), oae obtains

C X 1 - (5-256 C' JE'.'ls':. f .r r za

= j crslkz - z làq-m.I,tJrlrnp@ )2:%f g -a. -.

'))dz' dz (It + 2r5 J* (mz@))2 &. (7.3.57)- rutzlms (z

The azvanuge of tikis iateaal over oae witk z - z/ iaside a logarithmshonld l)e quite obvions. This erpressioa will be used in sectioa 8.1 for theealenlxtion of the mMnetostatic energy of one-dimemsîonal domain wallsin thia flmq.

8

ENERGY MINIOZATION

8.1 Bloch and Ndel Wnllq

The most popular case of minimizing all three energy terms (namely, theexchange, tke uisotropy and the magneiostatic eneagies) is the study ofthe stntdure and enerr of the wall between antiparazel domains in thinilms. The Izandau and T'A'RM#,z soludon desczibed iz gecion 7.2 assume anlnBn-&ta cnrstal, in which it ks possible to get away with no magnetostaticenerr contribzltion. If the crystal is lnite, this wall strudure contins anon-xro normal Ymponent of the magnetization on the surhce, and theenerr ofthe ensuing smvfnne cbargemust be taken 1to account. Moreover,NH Z'eI:O

- nlrpady in 1955 that the enerr of this sufMe charge canbe tx large in the case of very thin flms, which have more suzface thanvolume. Fbr tMs zeason, N&.I suggested a dferent structure for the wallin vet'y thin fzlms, in which the surface chaue is replaced by a volumeGarge, aud showed that the total enerr could indeed be reduced by sucha trxndormatkon.

This problem of a wall structure ia thin Elms will be dœcibed here forthe geometry showa in F$g. 8.1. A plate wltic,h is infnita in b0th the z- andz-drectkons has a thckness 25 in the tll-dz'wrta- on. Two autipxrallel domxinshave their magnethation along +z, which is alsoassume to be a.n emsy aisfor a unieal anisotzopy, aad the wa11 betwen the,m occupies the re#on-a S z K a. The wall is Msumed, ka tlu-K section, to be one-dimensional,namely m is assumed to be a function of z only.

F

b@ > XZ

'-'bea a

F1G- 8.1. The geometry of a domain wall ia thin îlms.

l58

One way to approach this proble.m is to n- the solution for the bnlkwall structm.e in eqn (7.2.17), and calculate its maRctostatic enera forthe'Ose of a fnite b shou ia Fig. 8.1. ()f coarse, it should be taken intoatcotmt tVt the bMllr Mrudare may be modifed for a fnite thicHess, andit is better to have a model with one or more paramcters and rnlnsm-tz,mthe total enera with respect to these pnmmeters. The model should onlytend to the structure of eqn (7.2.17) in the Brnlt b .-> x. However, thecalculation of this enera tel'm for tkis particnlnm wall stnlcture e%n only bedone by a Nlntvely complicted (292J numez'ical comptttation. Therefore,two methods have been used for resolving this dllcultjr. In one methodcer-x-n approimadozus for the maRetostatic enera are introduced, andthe other method uses flmctional forms forwhiG the mMnetostatic enerrrltn be calculated aualyticallyj and which appwxjmate eqn (7-2.1.7) for alarge 5. Examplœ of 170th methods eAn be found in the literature dted inlgxqâ aa.d (2923.

Here I choose to mustrate the problem by one of the models of thesecond type. lt was fa.s't propœed by Dietze and Thomas in a pape citedia (288) and (292), th= e-xtended to more adjustable parameters by others.The orîgînat paper is i.n Germaa, but. it is not imporuné for the reHe.rto look it up, because the calculation of the maRetostatic energy givealtere uses a completely diferent method from the one #ven there. Only theresult is the same. This model assumes that tàe z- and v-components ofthe lmlt vector m are

ENERGY MIMMIZATION

2 cos ('.iq .mwtzl = a ,tzo + ''z;

e sir/mvtz) = : + s , (8-:t.T)

wàere q is an adjustable paraeter, wllich œsentially determjnes the wallwidth. Here 4 ks aaother parameter which is introduced in order to treattogether the cases 4 = 0 (which mnlrfw my = 0) and 4 = xI2 (whichmakes mz = 0). nese cxq- were studied separately in the oenal paperof Dietze aad Thomas, as well in all other models of a one-dimeasionalwall- It should be noted that in thc case 4 = x/2 the volume cbxrge in thewall vaaishes, but there is a surface charge on v = +5. This cmse has been#ven the name of the BWh vpcll. On the other hand, in the case 4 = 0there is a volume eàarge in the walk and no sttrface chaœge. This case Lsexlled the N&l =cll.

For any value of $, the defin-ttion of the unit vector m is completH bythe requkements that m2 + m2 + a2 = lj and that at the end G the wall,> M z;where the domains be#n, mzt+xl = +1 (see Fig. 8.1). Heace,

m 2ç2 + z2'ma(z) = z z .

c + :r (8.1.2)

The maaetostatic enerr of this wall confguration cxn be calenlnkzvl 9om

159

eqn (7.3.57) for the Nrticular case a = x, which is implied by eqns (8.1.1)aud (8.1.2). For the ktegrations over z and W it ks only necessary to use

BLocH ANo NàBL wALLS

'x' statzj)& = 0:2 + z2-x q

E'* costràl 'r -.gz = -. e ,2 + ZQ g-x ç(8.1.3)

and-

q 4 T q z z z 'x' u,:J-- Lq, aazzlz

'* - -2 e + z2 + ca'dan -q --

- 2 ' (8.1.4)

Subsdtuting a1l these rodxuons in mn (7-3.57), the magnetostatic enezgyper nn't length in the z-drection is

ço .-24:su = zlrplMszçcosz 4+ 2x2Ms2g2(sin2 $ - cos2 $j J *

j (1 - c-2&) dt.J0

(8.1.5)The raaining integradon over t is a wemuown Laplace trausform,

whicà allows the whole expression to be wHtten in a.a analytic closed form.Howevery in the study of domain walls it is custome to deal with the wallenera per unit wall arew denoted bxg y, rather thau with eperr per unitwall lemgth. la the case of Fig. 8.1, it Ls necessary to didde tke FaE energyper unit length by the 61m thickness, 2% to obtain the eaera per unit wallarea. Theefore,

SM = .,2.v,2 qcozz 4 + -jQ (sin24 - cos2 4) 1% 1 +

-b 1 . (8.1.6)>= 25 q

ln particular, for the Bloch wa,ll wit: 4 = x/2, this enerr is proportîonal'k> (:X)log(1 + 5/:), wkich tends to zero for b -.+ x, aad rtmuà-nn im'tmfor 5 -+ 0. For the NIeI wallz wit,h $ = 0, the menetostatic Gergy isproportional to 1 - (:/$1og(1 + bjqj, whic.b tends to zero for b --y 0. andremains fnite for b -+ x. It Ls thus qualitatively clpxr that if there areno other types of wn.llq, the NV 5vall should est for thin Slms (h whichthe enerr of a surface càalze ks hrger than that of a volume chaœge) , andthe Blocà wall should take over for thic.k flmq, for which the energy of asnrfv- charge becomu smaller th= that of a volnme charge.

The other enera terms to be considered aTe tlle exc%xnge and theYsotropy. The energ density for the former is #ven by e1n (7.1.4), whkhbecomes, 0e.r substltuttg Fom eqns (8.1.1) and (8.1-2) azd cazrying outthe diferentiadons with respet to z,

1 dmw dmu dm.z 2Cqqzu = -C . + + = . , z a: ,2 dz & d:r Lq2 + z2 I2( 29 + z )

( 8 . 1. .7)

160 BNERGY MINIMIZATION

where C is the exchange constaqt defned in section 7.1. Note that thksApression is independent of the parameter $, which was entered in thefo= sin2 4 + cos2 4. The integration is obvious, and the (.= e energyper unit wall area of this model is

5 x c%x 1 g''yox = - = - m6dzdy = -(W2 - I).2à 25 -s -x c (8.1.8)

Aasnmnsng that the anisotropy is uniax-ialj whose e%y a'ds ks parallel tpzb the anlsotropy enera density is #ven by eqn (5.1.7)9 whicE will be usedhere for the case of a negzgible Xa. For the partknlnm consguration in eqn(8.1.1) this eaerr density is

42 2 q'tpu = Xh lmz ..h- ?'??,vl = A'z c z z -(q + :n ) (8.1.9)

SuVtituting in eqn (5.1.10), the anîsotropy enerU per unit wall area is

?lm 1 A'qw = - = = v?u &dv = v K: .

26 23 -, -w - 7 (8.1.10)

As was the cmse with the exchauge tprrn , this exprvion does not dependon $. Therdore, minimleing the total wall enerr with respKt to $ isMhieved by ' ' ' ' the magnetostatic term only. And since *'yM/:4 isproportional to sin 4cœ 4, there are only the two solutions mendoned intàe fore&ing: the Bloch wall, cos $ = 0, whick àas a sur.far,e charge but novolume charge and whose total wall enerr per unit wall area is

vcc x cz-vzzz'' b./j - 1) + OJt', + - s log 1 + - ,':&! = ( , ) q

(8.1.11)

au.d the Néel wall, sin $ = 0, whicA has a vohme charge but no surfacecbn.rge and whose total wv energy per unit wall area is

rr C : Jr....g c kzz, y . : s g ) .y. ( . ( a o1-yz ;..)x = (V% - 1zI + Si + c , q oq 2 t.l

Note that the exchaqge enerr term is trying to make the wall width, q,as large ms it can, while the Ysotropy enerr term is trying to make q mssmaz as it cltn . This tendency is more general than the partic'ular modeldiscussed here, aud f'ks tàe geneml, qualitative iscussion în section 6.2.1.The role of tàe ma&eutatic enerr term is less obvious because of thedependence oa the flm tMclmess, 2è. TMS feature is also rather typical,in that the magaetostatic energy term is usuatly yttite complicatM, and

BLocH AND NtBbwM,l,s l61

it is not easy to s- its tendency and preferences. J.n the present cmse, theonly obvious feature Ls that 7M prders a large qt if bIq is constaat. Buttlis statement is not helpful because b1q is not a constant. Even in tids

- . Mmple case, the only way to fnd out the role of efu is to ml'm'rnsze thetotal watl enerr for deerent values of Ms and try to see the tendency.nere is no mMnetic feld in this calculation, whic,h is just meant to fndthe static structure of a wall in zero applied Eeld. It is not dilcult to adda.n inlraction with a feld to this model) but the main efect of applyhg afeld is to make the wall move somewhere else, which is a diferent problem:

The parxmeter q is determined by minimizing the wall energy in eitherc'qn (8.1.11) or eqn (8.1.12). It is nrhieved by equating to zero the derivativeof the wall enera with respect to q, whicx len'q to the transrvmdentalequation

C (H - 1) = AK + A.u,z 2-1 1og (1 + ! ) - q

,n' 2 b q q + sq (8.1-13)

for the Bloch wu, and to

C (H - 1) = X-.-!- + g'u: 1 - 1 log (1 + -9) + q

, (8.1.14)'n'q 2 q q + ifor the N&l wall. These equations have to be solved for q as a faacdon ofb, and then the emc'pv r-qn l)e calculated from Gm (8.1.11) or mn (8.1.12)jby substituthg the computed q.

The solution of these equations is straightforwaxd only in the limit b -> Ofor the Niel wall, or b -.+ (x) for the 'Bloch wall. ln b0th thOe cases themaaetostatic energy Ontribution vanishes, and the solution of dthe,r eqn(8.1.13) or eqn (8-1-14) ks

2cq = v(W2 - 1)..tA l .

(8.1.15)

Substituting i'a mn (8-1.11) or eqn (8.1.12),

hm ,m = nm % = c 2crz(X2 - 1). (8.1.1 6)

This wall energy is 'a'tx/i - 1)/2 = 1.011 times the enera of the Landauand Lifshitz wall, eqn (7.2.18), which has b-n obtained as a solution of theEuler ewation of the problem. lt mpAns that eqn (7.2.18) is the absoluteminimum for the enerpr of all possible one-dimensional wv s'trucwturO in anklnite 5lm thiclmess. A diferemce of only 1% from this absolute mYmum,and /or any wal'ue o.f t?Ie physical pcrcmeïer4, certinly makes the pre-tmodel a very good approzmation, at least for very thick Kms. Also, the

162

1!l

$'

i rn15 ! 1 O--' ' Nëel ClF ï -'.x

a u:<v' S ' ;.7 C2tr N w

- - 4N .-. <' ** pwA , o .. ... ..-

, j 5121 iw ... ..- njoaty gj= ''v..x. c. ..

.-* w tzç7 --. .-. -. .> -! >r x ox-w u'-'< I c)

q q =5 .Z :ör- j >

rk!p. -

() I I I . I . .. . , 11 i. . ,. I

0 100 2OO 3O0

1020

ENERGY YY/AAON

Pilm thickness, 26 (nm)FIG. 8.2. The domaH wall width, q (dœshu curvœl, aad emergy per unit

wall arem J, for Bloch aad N&I walls iu thin permalloy S=s.

wall width whieh caa be deèned by q in eqn (8.1.16) is not signKexntlydxerent from the width obtized 1om eqn (7.2.17).

For any fnite Glm thintmœs >ns (8.1.13) and (8.1.14) have to be solvednnmerically, aad a nurneric,al solution can only be performed for sptmiicvaluœ of the physical parameters- As aa Awmple, I choœe the parametersusually used iu the study of pqrrnatloy êlmq, namely C = 2 x 10-6 erg/cjKï = 10D erg/cm3, and Ms = 800 emu. For thœe Guesj the compu%dvaluœ of the wall width paramde,r q aze the dashed line plotted ia Fig.8.2 as fnnctions of the fll= thieWess, 2è. Once q is Hown for either of thœewallsj its mlue rltm be substituted ia eqn (8.1-11) or (8.1.12), for compntingtàe wall ene-rgy per unit wall a2% 'ys oz erx respectivdy. The enerry valuesthus computed are plotted ms the 6111 cuzves in Fig. 8.2.

Deerent one-dimen/onal models were publisked, aad they atl yieldedvery Kt'=t'lar zesults) for the >me values of the physieal parameters. One ..wof the obvlous theoretical conclus-ions Fom Fig. 8-2, whie,h has alreadyben stated qualitatively in the foreming, îs that one should expect N&lwalls in vezy tkin Sms, aad the,n at a certain flm thinkncs there should

l63

be a shao tmndtion to Bloch walls. This sharp trusitions did not s-rnright, and several workers tried to work out a cerœn mted mall aroundtke transition be>een tàe Nlel and tàe Bloch wall re#ons- None of tàesemodels worked, and they a11 collapsed in the: same way that tke presentmodel in eqn (8.1-1) did. In tàe beoning, this model contained an Gtraparameter, $, wldreh could kave values for wàic.k the wall is partly N&land partly Bloch. However, tke enerr minimivnàion retained only tàe twoYues O and rj2, and did not allow aay mixlng. Tàe same àappened forany model wbbcà Mybody tzied. Later there was a general proof g%lo) thatthe same must àappen to any one-dimKsional model, and there c4.n be nomixed wall in one ds=ension. 'I'Ms tkYr= does not nerxqm'ly invalidatecertain semi-quantitative arguments (293) about the possibility of a mixetbut not strictly onmdimensîonak wG.

Experlmen#mlly, 't'àe transidon 9om a Nlel to a Blocà wall ls not sharp.It is prwmlble to disdnguisà - entally between tke wa11 structure intMn flms, identled as a Néelwall, and the wa11 structure in tkic.k fll'rns (orin bulk materials), identfed as a Bloch wall. Howevery between tàe re#onswhere one or tke other is observed, tàere Ls a ceexin reson of Alna thick-nessœ in wàicà a tàird type of wall is obsened. This third type, which hasbeen named tàe cm=-tie walb àms (52, 294) a very complicated structure.It is defaitely not a one-dimensional structure, b6x:a,11- it àas an obviousperiodicii'y in the z-direction of Fig. 8-1. Tàere àave been several attempts(295, 296, 297, 2981 to work out a theoretiv modeel for tàe magneœationstructure in this cross-tîewall, but none of them could produce xtisfactoryz'e-cnlts. More vent computations (299) made a large admace towards tàeunderstanding of this wall strudure, and compared hvourably (3œ) witàex-periment. However, they have not really solved tàis problem comple*ly,and the fne details of the cross-tie wall strudure are not very well kztownyet.

Tàe detxilK of tàe N&,I wa11 structure are not ve& Fe.II Hown either. In1965, Brown tried to avoid the choice among tàe large number of tàe tàen-efsiing models for the Bloch and N:el walts. He tàougàt tàat àe conld fndthe structure witk the lowest possible 0e.1v by a nnrnerîcal zniaimizationof all mssible one-dimensional co rations, usiug a method which willbe decribed in Gapter 11. He and Ms studct, Leonte, solved E301) thisproble,m for the Bloch wall in pe=nlloy Xlas, but tàey could not fndsuch a structure for tàe NGI wall, because tàe computations just did notconverge. bt turned out that the Néel wall àas a verylong 'tG', wàicb keeps

- g witk fartàer iteradons.

Later Omputations, reviewed in (295j, found various ad hoc solutionsfor tàe problem of cmvergence of tàe tail, but tàese solutions onlyproducea converglng result. Tàcy do not necœsxrlly solve the pàysical problem,and it is not cleaz at all if these converog solutions Rtua'lly present apàysically valid wall structuze- In tàe îrst placej the necessity for special

BLocH Axo N/;>:L wxtrs

164

tricks by itself should be regarded ms a smptom of a deeper problem, whichdoes aot go away wh% the emptom is removed. Secondly) details of thecomputed waz structure do not quite agree (302) with expeziment. Thirdly,the whole one-dimensional approach ls bmsed on the assnmption that mg iszero everywheze, even though the magneticîeld in this dirrtion, 8Uj%, isnot zero, whic.k somehow does not sound right, as noted (29% 295) already,togethc wit,h some other details. Besîdes, there is too big a diferencebetween the tails obtained for the same physical parameters, with a slightlydieerent flm thickness, as in curves c and b ia Fig. 3 of (3031, and it looksstrange at best. On top of that, there is a general theora (304) accordingto wkic,h alI one-dimensionk magnetization structuro are unstable.

It is quite pnqm-ble that these dl-lculties are not serious, aad ai leastthere is no expedmental or theoretical proof that something is baniœllywrongwith the theozy of the onedimensional NH walt. The theore,m aboutinstabihty of a1l one-dimendonal structures was neve,r tnkm very seriouslyby anybody, not even by the authors of the original paper. For the par-ticular case of tke Landau and Lifshi't,z one-d'zvnensional wall, they wrote(K4) tHt it tderive its justifcation Fom Shree-dimensional considerationsimplidt in the initial statement of the formally one-dlmensionat problem'.They bazl only a very mild critldsm of other wall Gculations. On p. 93of his book (1<j , Brown still justifM the Landau and Lifshitz wall cal-culations, but was more ex-plicit in stating that the calculadons of wallsin thin flms are essentially invalidated by this theorem. He wrote thatthey neeéed jnstiâcation, without wlzicà they fmust be regarded ms mereguesses'. Hoèever, this Gtidsm was Just Wored. Everybody else in thosedays regarded this theorem as a mere formality and a nuisance, and mostpeople thtnk so even today. They consider it az sometbing equivalent tothe mathematîcal proof that magnetism czmnot Hst iu two dimensioas,wkiclz does not prevent a theoretical study of two-dimendoaal systems,and ita comparison with experiments on nprly two-dMensional umplewms discussed in section 4.5. They believe that these one-dimensional Nëlwalls, aloough formally wrong, aze a vezy good approzmaiion for the realthreedMensional wall strudure- Therefore, no serioas attempt %M eve.rbeen made to check this point) and all that is known about NG.I walls hmsnot changed since the review (2951 which listed the best one-dimenm'onalmodels that werc all geaernlszations of eqn (8.1.1) with.more parameters.The results of aa attempt (294 to allow mv' to be a fttnction of yt instèad ofjust 0, were not very encoura#ag. 'ne most xcent numerical computations(299 302) nlm start with aa c priod assumpiion of dependence only on z!of Ftg. 8.1.

tn view of all tke indications mentioned here, EE do not consider thisapproach to be satisfadozy. Although there is no clear-cut evidence for it,the solution may be in allowing another dimensson. Since the strudure ofthe Goss-tie walls involves a handedness which changa periodicazly in z of

BNERGY MINMZATION

TWO-DIMBNSIONAL WALLS l65

Fig. 8.1, a real Néel wall may also itave this periodidty. This calculationhas not been tried yet.

8.2 Two-aimensional WallsThe theorem mentioned in the previous section, about the instability of allone-dimensional ferromaaetic coafguradons, also applies, in principle, tothe case of the Bloch wa11. Moreover, while the possibility of a-dependeacestated in the previous section is only a spemklation for the Néel wall, there isstrong experimental evidence (52, 305, 306, 307) that such apedodic ebxngeof the handedness dœs e-rut in Bloc.h walls, eve.n in bnllr œystals. Thereis also a rather convincing argument (295, 308) that this periodic changereduœs the wall energy, at least with respect to the onmdlmensional Blochwall. Still, the CFeCI of this z-dependence on the wall enerr, and on itsstructure in the othe dimeasions, has not been fu'tly investigated yet. Theusual assumptionl which has never beea justled in any way, is that thez-dependence îs a minor perturbaûon, whkh Oects only a small part of along wall, aud eltn be ignored without making a sezious mistake.

However, for the Bloch wall there was also a rather common fraollngthat tkere must a wa.y to reduce the magnetostatic enersy by allowing avaziation of the maaeœation along y of Fîg. 8.1. Brown in particular uscdto go around advertksing tkis idea, but neither he nor anybody else had agood modd to try it on. The frst published suggestion (dted in (2952) wasa perturbation sclzeme of the one-dimensional wall, which has never beenactually carried out. Then there were obsmatiomq of covzzled walls, wbichare the walls în a sample made of two fcromagnetic Alms, separated bya thîn layer of a non-magxetk materbl. There kq a strong magnetostatîcinteraction, eecting certah experimental reqlts, betwœn a wa11 in theupper layer wllic,h is Just above one in the lowc layer. Some rderencœ toboth experiments and theory are given in g295), but the detnsh are outsidethe seope cf thîs book. It is suEdent to mention he.re that the theozy ofthis phenomenon usd two-dsmensional models in which the magnetizationia the wxllq wa: a ftmction of b0th z and y. The purp- was to form closedJoöp.v of the magnetîzation vectoz, whiclz do not have a volume cbarge, andto hit the surfafes at small aagles, thus redudng the surfRe nhntge.

lt then occurred to me that the snme model, in the limit ofthe thiekmessof the separating layer tending to zero, ynn.y be used for the wall stntcturei.a one flm. The only diference between à, single wall and a pair of coupledwalls is that in the latter it is prv-ble to draw closed loops whose radiusshrlnlrs tozeroat the centa where there is no ferromagnetic matczial. Suchmall-radius loops involve a very large exchange energy in a single layer,whose eowtre is xlg.n ferromagnetic. 1 solved this diEculty by deêning a smalltransition region at the wall centre, Ic;l < zo, in which the magnetizatîonchanged more gradually than in the coupled walls, thus avoiding the largeexcAange eaer&. Tbis zn was left as a parameter with respect to which the

166

enezr was minirnTzed. Two more adjastable pnmvneters wea'e suae(:*9j, but never tried.

Splvn-: e.a,lly: for the geometry of Fig. 8.1 the model (309) %sumed thatfor IzI < zn,

ENERGY MTXIZATION

X#Tllx = - Sm -

N) ?

*7 o s ( X-F )mz = sin - c ,zzo 25

rcz c. (%; jm. = COS

22c 2à(8.2-17)

while for lz1 > z(b

mz = -secz gla, (Izr - zoj sin (:!J?) j ,

mv = jzzjsech gj (Iz$ - zclj txnh gsm (IzI - zaj cos (-7j) , (8.2.48)

zn. = .--jzj ttaaip (w* (IzI - zolj + seehi gwX (Izl - zpj c.c (-0 ) .z 25

Note that. This model obeys the constraint

'm,Z + m,2 + zn,y = 1 (8.2.19)I F

everywhere.* The magneœadon is condnuous everzwherc, including at I = ézo.* It represents a wall h the sense that mz = +1 for z = +x-* There is no suzface charge, becuse vd,./ = 0 for y = ::E:è, but there is

' a volume Garge. Thus, this model doe not go along witN the ruleargued in sedion 6.3: acmrdjng to which a xmrfltce charge should bepreferred over a vobme e in bulk materials.

The structure of this wall model ks shown in.Fig. 8.3, fortNe particular caeof a permalloy 61m whose thickneu is 2000 l., with the value of zo wkicllrniminnl'ze,s the emerg.g for these partictïlxr parameters. h this fkgure, onlym. and rz?,v are plottM. The component ma is perpendiciar to the plaaeof the plot, aad its magnitude is large where the size of tke plotted arrowsis smal), and vice versa.

The ortnal publkation (309) of this modeldid not contain this fgttz'e:because titis method of plotting magnetization structuro was only ilwentedlater: in the thesis of Leonte. It shows how the volume charge is decreasedby forming npxrly dtred loops, in whicà the head of eack arrow nearlyfollows the tail of the one before it. Thks Hnd of magaetl-don structure

TWO-DIMENSIONAL WALLS1 4 t t z .' ,, ,. .- - -e - y .. x x h : # 4 ,I 1 p /' J .e ,' .e ... ... ... %.. w x h. h v $ ! ! Il l l z' z z e .' .... ... - - ..w .% x q. hk & l t Il l i I : g' # z' .- ... ..- ... .h, .x x. x A N l ! fl l l l / z z .e' .e. ... - .-. x x. x 's N ! î t #j l ) / /' .' z ,.' .e, - - ... .... w x x N. ! l I #I 1 l I / /' z z ,,e .e. ... ..- .w x N h. h. !. $ I fl 1 l l l 7 z z .' ..- - -u. x x sk N h ï l 1 !l l ) i I / z ,' e .e. - '-- x x h'

'$' !'

! t t tt l J I 1 l i ? ' ' - - .. '$.

$'

l t t ! 1 @.-..1. -L..).--1-.-l--.2--I--' --...... -= ...- o-.t. -1 - -1--1--1-..1- J- -t--rIt$hNNNxx----zz,//Jll

zxalyqkxwxx- v-xzz//?,!) liïhk%uNx----zz////,!

.. . j- x ( . -( ... j ... p .u (.. ..j .. . j .. a- n .- - - r -j- -t **j ev j- *q.- *( e..'j. wt -- j-l r l t T ï k . . - - ' ' , J ? l l I t 11 k $ ï ï 1 '. '.. h. -- - 'e e .' ,' / ! t l t 1l ) ï '$ 'k. N N .k % w - .-e .,' ' r ! ? ? 1 @ tI l ï ï N 'k s. x -.- - - .-- ,,' .e' z z' ! t / t lI l $ '$ '$ N ''. N -.. -- - -'- .'' .,' ..' z' t t l ! 1) t $ hk N 'x N Nh N. -- -'' '-- .,' e' .//' ' /' / ? f !1 $ ï N N N x 'hs -... .w - ..- ,,' e' e' ./' z / / f ti i h N N N %. 'h 'x ... ..p. .,e ..' .e ,: #' z : # l Il I I . N l x. w h. .. *. ,. . . 1 e e / I z pi l A Ai % h %. h. .h. .. .@. * .- .r .e'

e' e'

a 2 # I

- ..... - z N.apca 2 . .

167

lxFlc. 8.3. The assumed structure of the flrs't two-dimensional domain wall

in permalloy El=s ms described by eqns (8.2.18) and (8.2.19). '

is preferred by the magnetosutic energy, but involves work wlkich mustbe done against the exchange enera, which prefers the magnetization tobe aligned. It is especially noted in the fgure that if the region with zoLs removed, a circular vortex with a very small radius is formed at thecentre, which is better avoided because it involvœ a very large exchangeenera. The introduction of this extra kansition region ks an artifdal, adJztpc solution originating from the adoption of a model for double walls,wkth no exchange at the centre. The physical system can fnd better waysto avoid this large exchange at the centre.

The ex and anisokopy ener#es of this model were calculatedanalytically. The magnetostatic enera term was expressed (309) as a one-dimensional inteval, which would be trivial to compute nowadays. How-ever, computers in those days were not what they are now, and instead ofcomputing this integral, it vxks only proved that the term whic.h contaînedit was nenative- Therefore, by droppîng that term the magnetosVtic energywas increased and the totzal wall enerpr thus calcuhted was an 'apper b/undto the wall energy which can be obtained by such a wa!l structure. Suchan upper bound was adequate to demonstrate the necessîty of this seconddimension, because even the upper bound for this wall enera was alreadyconsiderably smaller than the computed g301) lowcwst possible enera for a

168

onedimensional wall.Ia a way) this demonstration was a waste, because by the time R was

published LaBonte waz already concluding his doctoral tbuis, in whieh hedeveloped a method (to be desebed in ehapter 11) for numerical Tn'-m-t-

miem.tion of the total wall enerc, composed of the exchange, auisotropy,aad magnetostadc terms, of two-dt-melsional magnetization condgtzrations.He computed the strudure and enera of twœdt-menional walls by this nu-merical minîmization, which was muc.h better tba.n this crude mMel- Theolly usefnlness of the model was that it gave Hubert the idea (248) to con-struct twoemensional *a11 skucture with zezo magnetostatic energy. Heintroduced the constraints that mg = O oa y = Hzà of Fig. 8.1 arltl that

ENBRGY MMMWATION

num am v+- =0dz oy (8.2.20)

evezywhere. 'nese conditions were Kforced by choosing a scalaz functionA@, y), with .A = const on y = +b, and defning the components of m by

tz?.4@,:) aA>,y) a a.al)ru = , mv = t-s , ( -

%with m,z beingdeîned bythe constrintofeqn (8.2.1S). To simplià matters,Hubert 2248, 24% ckose a eertain fundional form for X@,:), which con-tained certain adlustable htnctions. Thœe fundions were eveûtually defnednumerically fz'oz poiat to point dadng the enerr minimization process.

For his doctoral thois, lmRnnte solved the problem of the Bloch wall i.npermalloyflmq, but at that time hewasstill usingsymmetry considezationsto reduce the computation time. Hc actually computed only the quarterz > 0 and y > 0 of the equivalent of Fig. 8.3, assmming that the restof the wall can be obtained by Gklng the m'lrrfar images of this quarter,which Ls the same mssnmpdon used to make the model skown izz Fig. 8.3.However, when he graduated, and went to work for CDC compans he hadunli=ited computer time at Ms disposal, which was umlsmnl in those days.Tberefore, he allowed the computer to lookat all four quarters çtî the wall,aad found (310) tha.t the wall enerr could be vezy muck reduced by astructure wllie,à is rptë symmekic along z. When viewed fzom the domaiaon it,s right', the wall lxks deerent iaa when viewed fzom the domainon its left. This result was unexpected, because one would tend to mssnmethat tEere is no built-in directionatits aud the wall caanot p=ibly teltwhich is right and whic,h is left. Eowevez, it twrns out that the azmmetricstructure allows the magnetizaïon to build nearly complete vortices, witha very small mn.aetosutic eaerr, 'tvitho'ui mltksng the 6.11 drcle at thecentre, with its large exckauge eneY. It is a better sohtion than the odltoc avoidaace of that small circle by the tpo intrMuce' d into Fig. 8.3.

The approHmation of zero menetostatic enera used by Hubert also

TWO-DIMENSIONAL WALLS l69

1ed (248, 249) to an asymmetric wall, and its whole structure tllrned outto be very Mmilar to that computed by LaBonte. The tosal waz energyVculated by these two methods wms also ver.v nearly the same, whickshows that mlnt*mlAing only the magnetostatic energy, as done by Eubert,is a good approvlmniion to minMizing the total wall energy, as done byLeonte. This result demonstrates that the magnetostatic enerr is theleading enerr term in suëciently largc samples, which is the conclustonalready reached in section 6.2-2- It should be emphmsized again that itis mnsnly the magnetœtatk enerr term which determines the complexskucture of the mMnetizadon in the wall (or ia any other magnetizationslzucture iu bnllr materials), while the exchange a,nd azisotropy energyterms only play the tole of small puurbations-However, onœ this stntctureis determined, the magnetostatic enera term computed from it becomevery small. In the computations of Leonte (310J for permalloy mms in thethicknœs ramge between 1000 and 2000 lo the ma&etostatic tqnn =iedbetween 5 % and 3 % of the total wall enegy. For the thinnc flm witha thiclmess of 500 â, the contribution of tMs tqnn was 12 % but at thisthicHess tàe minimization of the Bloch wall becoma doubtfal, becauseexperiments show that the crosmtie wall alteady tn.kes cver- Experimentson M platelets show (311) a strong dependence on z of Fig. 8.1 in the BlocEwall, already in the tàicHess range approvhing the occurraœ of Goss-tiewalls, and an even more complu tansition has been observed (3121 in aaFNAI alloy. lt is reasonable to assume that this z-depeadeace also afcmtsthe z- and v-structure of the wall, aad probably its energy too, in tàatthickness re#on, but the appropriate theoc has not been worked out. Htho reson of 11l= thinltmv it thus seems.that the calculations of LaBonteand of Hubert ars unreliable, and should be replaced by something else, inth- dimensioas, wbicE is not known yet.

For thickcr ilrnsv the azymmetric two-dimensional wall as computed byLeonteor Hubet ks h good agrYment with electron Mcroscope studies ofthese w=llq, at least to within the Mcuracy of these exeperiments. Actuazy,some wall asmmetr.y e.xn alteady be seen in older pictures (3134 whichwere published before these theories, but at that time tkis asrmmeky wasignored. When mote atïeation was paid to this detail, a vez'y pronounced

etry, qutte sîmoar to the one predicted by the theou: wms seen 5.a180* walls (314, 3151 316) in vazious matedals. Such an asymmetry hasalso been scen (3171 în 90Q domain walk, wbicE are outdde the scope ofthis book. :(n these measurements electrons are shot throngh the 61-, aadmeasure only the average of the maaetization in their path, namely theaverage along y of Fig. 8.1. However, in some cases several pazsœ are tn.kenwith ihe sample tilted (318, 319) at deerent azgles. This lozwhnx-que allowsa better l*k into the v-dependence of the magnetization, because severaldxerent averages a're measared. The azcttracy is not blgh, but to within thisaccuracy it seems that the computations of Eubert or EaBonte #ve a good

170

description of the obset'vu walls in this iatermediate Elm thinknus. It isnot sofor the thianer 61nu, wherethe theory does not ât the experirnent, ashms already bean mendoned. Apd it enmnot be cll<vtked for very thick flms,thtough whic.h eledroms emannot pmss without beg completely absorbedon the way. It may be worth meatiomfng that while enrth wall in a flm ismspnmetric, the nmple as a whole does not have any Oectionality for thisazymmetry, and walls with an opposite s-nne of asymmetr.y =ur in thesame Kxmple. Sometimes the sense of msymmetry rAn be seem (320) to vatyperiodicazly in z even within the same wall.

J.n as much ms a plot of nnmeriY restllts can be taken ms aânal solution,the wall structure in this intermediate âlm thirlrness is knowm Eowever,numerîcal restklts apply only to the particuln.r physical parameters USH fœtàe computations, aad if one wants to know the structure for a z'Yea'emtex constant, or Jmiruatropy constant, etco the whole computationmust be rexated. Alqo, dd!#m.ilK of the re-sults cannot be published, andMnnot be pmssed over to somebody who wants to use them ms a start foranother calculation, or to calculate a propeo whicx %M not beem includediu the original œmputation. For such purposes it is much more convenientto have aû andytic cmrorimcen of the magnetization structure withsome parameters that rm.n be ftted to aAy Nticnlnr cae. When a goodapprofmation for the true minimal eme,rv state can be spcifed over awiderangeof the physical parametersj it ks suEdent to m-tniml'qe the enerawith rapec't to the adjustable parameters for every spedfc caae. lt helps ifthe latter miaimization Ls simpler thaa the original nnmerical computatonof the whole strudure. But evea if it is not, it %M at lemst the advantage ofthe possibility of s

' '

g all the de' tails of the structure for any pazticularcase i:a terms of the numerical Oues of a fcxw parameters. Moreoverj itis possible to interpolate these parameters bdween computH values fora physîcal parameter (e.g. the amisotropy xnstant) and thus know theapplovlmate structure of a wil for cmses which have not been computed.

For the two-dimensional Blocx watl, tbere is su% a modd (321)1 witheight custable parameters, which is a very good appremation to b0ththe stntcture and the enerr of Bloch walls ms computed b.r TmRonte. lt istoo complu for any analytic calculation of any of the enera terms, andthe enera minimiRation must be done by the s=e numezical method asthat of Leonte (310$ but the possibility of rnmrnunicating the results interms of the natmerical values of the eight pnmmeters is a large advutage-Attempts to make a yirrlpîc'r model have concentrated on r-qAras for wkichthe magnetostatic euergy e.stn l)e evaluated analytically, because it is thisenerr t-rrn wMch takes Jtlmost all the compuiational time in numericalminîmlzations. h the best of tMs class of models (322), the eeuatâon.ofthe uchange and the anisotropy terms was done by a numerical htegrationof a one-dimensional integral for each. Such iategration is a relaively emsycomputadon, aûd the whole model is relatively simple to use, even if the

ENERGY MTNTMTZATION

TWO-DIMENSIONAL WALLS

defnitions of the magnetkation components take many lines. This modeltnrned out (3224 to be a suRdently good approxsmation for compadng withelectron microscopy data: but was not good enough for obtaining the Oerdetails of the theoretical t'wo-dimensional walls.

8.2.1 Bnlk MaterialsAbove a certpin fllm tlzicknessy even the highest-voltage electrons cannotpenetrate through the sample, and there is no way of kaowing what thedomain wall looks like. It is possible to shoot neutrons through the sample?but the accuracy of neutron doaction is just suëcient to see the domains,not the details of the walls.

Ixk the theoretical models for which the n.nîsotropy enera term can becalculated analytically (309, 322), this enera term inareases with increastg61m thicHess.' This term Ls negligible for permalloy 11lx'n whose thickness isaround 103 â for which most of the studies have been carried out. Howeverjfor a much larger tàickness this increase *1 make the n.nl'sotropy termlarger than the other terms, and the total wall energy will start (30% 322)to increase with increasing Gickness. Computations based on the modelof Hubert also show (315) tàat, at least in one cmse, the total wall energypasses through a rninimum, then starts to increase with tcremsing 61mthicaess. It seems that it is going the way of the aaalytic models, namelythe wall enerar wi'll keep increasing wità increasing 6lm' thicHess. Sincethe one-dimensional Bloch wall enera plotted in Fig. 8.2 decreases withGcreasing lllTm thickness) there must be a certain thickness above whichthe eneror of the two-dimensional wall 'rûl become larger than that of theone-dimensional wall. Therefore, the two-dimensional Bloch walls describedhere must cease to exist, and change into something else, above a certa'inflm thickness, both for a cubic and for a uniatal material-

For a long time it was taken for granted that at a suldently largethickness the wall will change into the one-dimensional Bloch wall of sec-tion 8.1) which eventually tends to the Landau and Likhitz wall of section7.2 in the bulk. For this reason the model of Jakubovics (321) was specii-cally designed to contain the one-dimensional Bloch wall as a particularcase for certain values of the parameters, mnlcing sure that if there is atransition to this wall it will come out of the computations. Such a transi-tion *om the two-dimensional wall to the one-dimensional one wms actuallycomputed (3214 for a large increase of the anisotropy constaat. It could notbe calculated for an increase in the 6J.m thicHess, because the require-ment of computer time and memory increases very rapidly with increasingfllm thickness, and a11 the computer rcsources are used up before such atransition is eve,n approached.

The same diëculty of limited computer time and memory also appliesto Leonte-type computations. The thickest flms studied theoretically sofar (323) are iron f 1ms a f ew ynk thick. H them the wali structure is still

172

predomin=tly that of the tkin flms? with no way of seng any posdbletransition tcwards a one-dimensional strudure.Near the edgedf thestudiedregion there seems to be (323) a dfereat ldnd of asymmetry, which maytake over at a stlll larger dlm thickaess, but it s'titl ha.q a slightly lzigheenergy than thai of the Lvonte-type thin mTn structure. Actually, theenerr dference between these i'wo s'tructurœ Ls so small that the computercan be siuckfor ever in the higher-eaenn siate when the computatioxs start1om kt At this stage it Ss not clear whether this other wall structure isinde going 1.o take over at ptal' 1 lazger flm thickaesses and whether it willeventually develop into something simzar to the Landau and Lifshitz wallyor ënto somethhg completely diferent. Some computations (324) were aISOcarried out for 10 itm thick ppvrnnnoy Elms, but t*ey used a very rough grid,with the subdivision being an ordez of magnitude larger thltn in muyothercomputatlons. Therefore, tke results of these computations aze unreliable.Budes, they yield (324) a maretœtatic emergy tezm which is 21 % of thetotaz wall enerr. It is suspiciously larger than ia maay of the other two-dimensional wall computations, aad seems toindicate axk hadequateeneramsnimiqation.

By analysing the polarizatioa of the electrons in a srztnn'mg electronmicroscope (55) lt is possible to memsure the magnetization of the l%ifew atomic layers near the surlce. Such expevimental data clevly show(325, 326) that near the surface the domain wa2 looks like a Néxel wall,in the sgmqe that the magndization there ks nearly pnmllel to the sarfMe.

'

risin because in ihe todsmensional will discussedThis ruult is not surp g,in the foregohg' tEe magneMzation also approaches the surface at a verysmall angle (see Fig. 8.3). At any ratej it should be dear to the reader bynow khxt tke mavetostatic emerr will not allow any othe,r approach to thesurface, even if the crystal is vezy large and the vast malority of the spirsare very faz 1om the surface. The working hypothus in Malysing suchdata 1326) is that ia a suëciOtly thick saple the wall is essemtially theoledimensional Bloch wall throulou.t most of the tMckness, but when itapproaches the surface it changu fzom the Bloch type to the N&1 type,when the magnetizatioa slowly tnrns around âom the v- to the z-directionof Fig. 8.1. Detaile LaBonte-oe computations (32% 327) both for ironand for permalloy support this pîctux, and are in good agrement withmeasured details of t:e surface part of the wall. They abo provethat cll theolder m-urements of domain wall nidtlt ia bulk materials have measuredonly the width of the surface part of the wall, whic.h is vez.y dlFerent fromthe wall width in the bul.k of the material. Eo-ver, such Omputadonsaze limite to rplxtively thin lms, because of limited computer resources,azready mentionM in the foregoing. The problem of what the wall reallylooks IiP.e inside bulk materials has not really been solved yet.

The wazl eaerr in bulk materlnls is not known either. The Landau andLifshitz result of Kdion 7.2 is still ofte,n used for analyshg domaln coafg-

ENERGY MYNIZATION

BROWN'S STATIC EQUATIONS

urations, but it is aot clear i.f it is a good approximation. By using a certainanalysis (32% 329), wall energies can be obtained from experimental dataon domain widths, and this memsurement is even often used for evaluatingthe exchange constant of the material However, LaBonte-typc computa-tions for thin flms involve no approfmation, except for leaving out thethird dlmension, and must therefore be at least a rdiable upper bound tothc real wall energy. And yet, wall energies measured by thss techniquef or thin permalloy 61ms are (270) considerably lcrlcr than this theoreticalupper bound. This discrepancy has never been accouated for, and it doesseem to indicate tkat something is wrong with the analysis of the data inthis technique, and casts some doubts on the values published for the bulk.

Finazly, it will only be remarked that there is a vast experimental andtheoretical literature on walls which are not straight lines, in particularwalls in the fol.m of a closed drcle, around a circular domaln., known ms ab'abble domcïrl. For such a circle, even a one-dimensional wall such as theone in section 8.1 must be expressed in the two dimensions of the drcle,and a two-dimensional wa2l as in (330) is too complicated to be discussedin tbis book. Some references cxn be found in (2882.8.3 Brown's Static EquationsNumerical computations as described in the previous section are relativelynew, and aJe still limited to relatively simple cases. Wit,h present computersit is not even possible to :nd the lowest-enerr confguration of a singlewall, let alone a whole structure of domains separated by wallsz or any othertrue three-dimensional magnetization distribution. For such problems it isstill necessary to look for aaalytic solutions, or at least worHble models.In as much aa the wall energy is Howa, it is possible to compare thetotal enerr of certain domain confgurations, and fnd the one which hasthe lowest energy. This technique is the bmsis for what is known as thedomain theory, which hms been used successfully for many cases. However,in principle it has two serious drawbacks.

The frst one is that comparing the ener#es of difereat conîgurationsalways carries the risk of ignoring another conîguration, which may have astill lower enerr than a11 the ones being considered. If the basic structuzecan be taken from experiment, or if it is done by somebody with a highphysical intuition, it may work out But the probability of a correct guessis never very high, and many wrong results have been obtained by thismethod. It has already been seen in the previous sections how all sorts ofone-dimensional models were compazed wit,h each other, till it turned outthat the wall energy èa'a be very much reduced when a variation is allowedin the second dimension. Many other examples also est, and it is alwaysa risk which must be borne in mind. J.n principle, a'n.y enerr calculatedfor àny particular model, with or without minsmization wit,h respect tosome adjustable parameters, should be regarded as an 'tzppcr bound for the

1;4

actual ene-rgy mMmum. We mMmum cxnnot be larger than the energyof any special oase, but tNere is always the possibility iat the lowest-encgy mlnsmum Ls in a dferent co tion, whic.N is not included inthe assumed model. The estimate is 'really rliable oaly if a Iower bunde%n AIM be found, in which emr.p the tzue mfnlrnum must be between thosebounds.

The second dMculty is that one is not alkays ia*res'ted ia the lowœt-energy state, because of the hrstereis which is part of the study of ferro-ma>ets. As *n atready in the simple czse discussed iu sectioa 5.4, theactual masnefsexuon state may depend on the history of the applied feld,and even though lower-enera states may >='qt, they aa inacessible dueto an enerr barrier betwe% them and the praent state. IIk such caaes,comparing ener#es is meaaingles.

Rr thue renArms Brown set out to eoress the energy minîmlxationrigorously, with am eye to perfovl'ng it in a way tàat would tn.ke the hys-teresis into account. A ftrst step iu this d''rt?M' on A'n.q already been demon-strated izt the Landau and Lifshitz wall in section 7.% where t;e enerrminivnsvlttion is done by solving the Euler dlWerential equation which Ieadsto tàe lowot possible ene,zv minimum for the assumed form of the to-tal e,neru. h that c%e there waE no hystereis, because no magneKc âeldwas allowed, and the eistence of the wall was assmmèd, c yritvi Brown'sidea was to bave the most goeral Euler df erentM equation by a pure=iational Xculation, so that the nviKtence of the wall (or of any otàermagnetization confguration) would be the ze,sdt of t;e calmzlation,witbouthaving to asstkme it beforehand. It was this theory that Brown o ' '

ynxmed nbicxmagneiica, althougll the name wa.s Iater exiended to meanany sort of calculadon in whic.N the atomic strueture of matter Ls i'gnoredand the magnedzation vector Ls taken as a continuoas ftmction of spRe.

Consider, iezefore, a ferromagnetic body of any shape, in which themaRetizadon is any fnnction of space. The total energy for tMs particularm(r) is male up of the cxcAange eneza as ia eqn (7.1.4), the anisotropyenergs the magnetostatic enerr ms in eqn (7.3.26), ald al interadionwith an applied magnetic îeld, Hy, wMch ks sometimts called the Z-maaenera term, namely

EIVERGY MTNTMTZATION

s - ,-+s-+. +su - / (vc g(v-.)- + (v-,)- + tv--laj +w

1 ,- -M . n - M . m dr + ws ds,2

where tàe last term of the volnrne integral is the hteraction of M with H.'as it *mu outof the denition of M, and as used many times iu this book,although not in this form. The ftrst integral is overthe ferromaaetic bods

(8.3.22)

BROWN'S AATIC EQUATIONS 175

and the second one is over itz surXe.' Both the volume aud the surfaceAnl-qotropy ene.rgy deasity are left unspeced at tàis stee, but they aresimple functions, which eztn be one or more of the cases speced ia section5.1. Magnetostrictioa r-stn also be added, but it is neglected in this book,as mentioncd in section 5.1: Gcept for cxses which eztn be written in t:eform of an anisotropy term aud are theefore induded in 'ttu.

This Gpression determines the e,ae,.r& if m(r) ks Howm The problemhere is to determiaè mtr) so that this enerr is a min'lmnm. Brown (14.51mlmlrnlqed tMs eme'ra iu several ways, the m-mplut of which is to consider amall variation of the magnetizxtîon vector around its value m:, bound bythe constraint that 1he ma>tude of m must be 1. The frst two Cartesiancxvdsnata can thea be expressed as

(0) + e'u mv = zn,(0)'+ 6v, (8.3.23)mx = rtm , y

where 'tz and 'p a.re any Nnctions of spaœ, and e is lImnll. The third com-ponent is determined by the constrrdmt that m is a Alnit vector. To a frtorder ia e, it can be wzitten ms

/1 . . z '.'l :c,a - jf' l - Lmzïz; + ,,tzl - Lmçlb' + cm) = y

mato - 2c tma$ol.?, + mvtglo.)(0) (0)(o) mu v + wzv k? (o) s a z4;= mz 1 - e : = m. - eA: ( . .

(0)mz

where(01 (0)'r4x u + mv t?l = . (8.3.25)(())

. TN.z .

The vazition of the exchange eaergy term due to this xriadon of m is

&s. - f / ( gv (mp) + cu)J2 + gv (,p,r + ,v?)j2 + gv (,pp) - eA)j22

g))2 - (v,p,;n))2 - (v.r4c))2) dr = .cj ( (vz,d)) . (vu)- tva+ tvmioll . (vz?) - (v,qo)) . (và)) d.r , (8.3.26)

to a frst order in T. However, arxmding to the divergeace theorem, for anytwo functions J aad F,

J(VJ).(VF) dr = j (V . (FVJ) - FVM dv = jFxygds-jnzfdp(8.3.27)

1V6 ENERGY MINIMCATION

where n is the normal to the snrfa-. Using tkis rehtion th'ee times in eqn(8.3.26) for the three prodttcts of tàis form whick ocftnr there, the variationof the excxaage enerr becomu

(0) (0) (0%. /0) rQ)ow t'gmv mx 'u + rzj n tqznhJ& = c C o, - + w ' - dstî'rz Dn zngol t':hz

( Z l ? 0 1 -1

c .p,v2m(0l + 'pV2.m,(2) - ---..-.1.--t.....-1 V 2 snYl o:y . (g.g.28;-E m p (o) .mz

For the oriation of the Misotropy energy it is suEcient at this stageto use eqn (8.3.24) in a ftst-ozde,r Taylor expansion,

8% = J (tu(m) - watzzzcll dr

(0) (0)t'guu f'l'tn rrzx .tz + my 'p dwa . := c u - + 'g - -- dzj (8.3.29:(0) iyvast a(e) drntolPms r z o.

and m'nn-larly for the mlrfAzte anisotzopy term.The =iation of the magnetostatic energy term 5s a Jr'iori #ven by

JfM ='-) J ((M + :M) - (H' + JH?) - M . H?) dr, (8.3.30)

where $Bl Ls the feld due to the small ma>etization vaziation z-M. If thisfeld had to be calculated by any of the methods used to calculate mag-netostatic felds, thiK problem would have beeome hopelessly complicated.Hcwever, it ic nof necessary to calculate this feld in order to eviuate theîntegzal in eqn (8.3.30), because it is posslble to use the recïprocilp tlseocmof eqn (7.3.40), which en-qlzres ihat in the pruent case

J M ' /$'N'' dr = J H/ - J'Md'n (8.3.31)

Substituting in mn (8-3.30)1 and leaving out the eeztmiorder term,

f:M = - ïT'.&T&dr, (8.3.32)

which has thc sxme form ms the vaziation of the interaction Mrith the appliedfeld, Hk. Using for both these terms the specifc wiation in eqns (8-3-23)and (8.3.24), one obtains

BRDWN'S AATIC EQUATIONS

(o) (:) ,h Fow 11 V 'm.y *é (SM + SsL, = -egzfs Szzu + II- :)a?; - Hz - ,o; dw, (8.3.33)>1

whereH = J:'f. + W. (8.3.34)

At a ndnlmnm, the variadon çtî the total e'nez'r, comprising all theabovomentioned terms, should vaaish for any càpice of 'u aud 4J. Thisrmuirement means that the coedents of 1ù and 't) in the volume intevalshould each mnxs' lb and the ume applies to the coecients in the surfaceintegral. Adding up all the appropriate terms, and omitting the iadex C0' '

which is not necessary any more: leads to two diFerentird muations in theferromaaetic body, and to two bozmdar.y condidons on its surface. Theboundary conditions on the surface aa'e

&mx znx (imz Dzu. ru 8%%5 . . - + .. = gdn m.z t'??z ' 0mz mz &rtz ' (8-3.35)

andêm v ma.z pm z (%p s m..z 8 t? , ôC ... + ... = .8n mz on 0mu mx 0mx

Tke two diferential equaîons are

mz a mw W'a TrG DnlxC :72 sa - - V m z + M s Sz - -Hz - + - = Oz

az my yrzw mz ômzJ. (g.z.z.jq

=d

(8.3.3 6)

mu c m. P*7x rzv omu6' V2m - T mz + Ms Fr - m - + = 0.& ma ma ômv mz xa

(8.3.38)It looœ as if mz plays a spedal role here, unlilce mx or mv, but it is oalya mattcr of cltoiœ whic.N two of the three components to use frst in eqn(8.3-23). The symmetry rxn be seen if cqn (8.3.37) is multipned by mvand subtrvted from eqn (8.3.38) multiplied by m=, which. leads to a thirdequadon of the same form:

Aa t'hrxc (mrvzmz - m.v2mv) +M, (mv&- nvamj-mvovv +mx = 0.. &mv

(8.3.39)These three equations can be writtea Ygether, in vector notation, ms

Dmzbm x CV2m + MsR - = 0, (8.3.40)êm

178

where ('ea/dm is a notation for a vector whose Cartesian coordinates are

ptna/dgrya, omulomu aad omwlomp. The vector notation is eazier for trans-forming into other coorinate systems, but it is somewhat misleadinp'ltshould be remembered that there are only two independent equatioms, andtke third one is only a linear combination of the other two, due to theconstmint (mI = 1.

These Guations are known as Brown's diferential equations. Theymeazh as phraed in (331), that in equilibzium the torque Ls zero every-wheare, and that the magnetization is parallel to aa eeetive feld,

RNRRGY OIMXZATION

C z 1 u&vHos = -V m+H - - .Ms Ms dm (8-3-41)

Since M x M = 0) any arbitrary vector propordonxl to M A'xy be Mded toHe.e without Ganging tàe rcult. In pxrfa-nnlnr (:t311, there is no dxerencebetween ushg E and B = H + 4rM.

Brou's equations have to be solved together with solving for H#, whic,llis pazt of eqn (2.3.34), by soleg eithc the diferentixl Guations in sedion6.1 or the ttegrals in section 6.3. The.solutioas of the whole set containia prtdple a11 possible enerr

' * but not only the mimimn- Thecondition that the vadationvaxishes Ls also Rtlfmed %remera mpm-ma, aadit is ne-sary to clzMk enztll soludon for being a memum or a minimum.

There pm xlg.n the boundary conitions of eqns (8.3.35) and (8.3.36))for which a liaexr œmbination rmM be added in the same way as to theHiierential equahons. A11 thre equations can then be writteu in a similarvedor notation as

m x C + = 0,'t%- dm (8.3.42)

on the surface- In the particnln'r cxqe when the surface energy is as in eqn(5.1.11)) nrkmely if

1 ,'tt)s = yff, (n . m) , (8.3.43)

one should substitute in eqn (8.3.42)o'w.

= Ks (n . m) no('?m (8.3-44)

whick is the form used by Brown (145), aad by others (332). Other, specinlcases, such as (146)) wîll be ignored kere. If there is no surhce anisotropy,which is the assumption made in most of the theoretiW calculations, thecombination of eqn (8.3.42) with the édentity m - 8nïl8n = 0, which holdsfor auy vector whose magaitude is constant, leads to Xnlnn = 0.

SELF-CONSISTENCY l79

8.4 Sez'-rxm6kx-<oneSolving Brcwn's muations is aot exy, but there are certain conclnsionsabout the nature of the solutions whick can be drawn right away. Forpvxmple, consider the ca:e mz = 0 or mu = 0, whick is the basic aumptionused ia all the one-dimensional wall calcnlauons in section 8.1. Substitutemv = O in eqn (8.3.38) yields E'v = 0, because cvery one of the othertezms vanishes, including onlxlnmv, which is also proportional to m.y forall the expressions of 'tra. Ia the absence of an appëed f dd, as in the one-dimensional wall calcnlxdons, Hg = 0 means Gat Hu' = % whicà is n:t' thecase in those studie that take the average of Hy' . Therefore, the Msumptionrzv = 0 (and s'-milxrly for mz = 0) cannot lead to a soludon of Bzown'sequatioms, and rlmaot represeat a true eneru nzinirnlTrn.

TV argument is rigorous, but it is not very useful because it cannotlbe mHe qnxnlaative. The fact that a partkula.r functton is not aa eactrepresentation of the actual physical state rAn hna'dly ever be a good rea-son to avoid it. Certain approimations are often inevitable) aad Oer allsome approx-imations have already been mGe on the way before derivingBrown's equafons. The real quetion is not whether m = 0 (or any other?assumption of khks sort) can be an absolu* enerr m-tnl'm um, but whetherit may be a reasonably good approtmatioa to this m-xnlmum.. The aboveargument does not help answe,r this quœtion, whc.h reqm-nps somemeasuzefor how far the model Ls fzom the mirlmum.

A partial answer cxn be obtm-nu when eqa (8.3.37) is multiplied by Tru,aad eqn (8.3.38) is muldplie by my, and they are then added together andintegrated over the volnme of the ferromaaetic body. Part of the kltegr=drAn be transformed by the relation

1 a 2 sa z 2m . V2m = -V ,rn .- (Vmz,I - (Vmp) - (Vzp,z) ;2 ' (8-4.45)

in whicà the frst term vaaishes as the derivative of a constant, aad the rœtis proportional to t:e exchange enerr densîty. The total inteval is thea

t'hu 1 ç'hvx z- zék + M . H - m . + - CV mz - MZHZ dr = 0.dm mz Dm.(8.4.46)

Comparing with mn (8.3.22), it ks seen that the total energy of the systemmay also be written as

1 okcu:7 = - 2'*a - m ' - M ' H. dr + ws &V2 dzzl

4. 1 aultv z+ - . - c 77 r'n z - Ms Nz d'r.2 znz Dmz (8.4-47)

180

This energy expression contains only a simphfed form of the exchangeenergy, aûd only that part of the magnetostatic energy which is included inHz of the last term of eqn (8.4.47). It is tlms much dum-rar 1.o compu'te tbanthe inteval in eqn (8.3.22). However, it cannot be used as a substitute foreqn (8.3-22), because kt applies only k) magneeation structure whicb fnl6lBrown's equations. What it can be used for ks as a mexsure of how dose aparticular model, or a particnlar minimizadon under constrets, is to thetrue energy mYmum whic,h is a solution of Brown's equationg. For a goodmodel, the energy ccmputed lomeqn (8.4.47) must be appremately equalto that computed from eqn (8.3.22). If these ener#es are vea'y dlFerent,the model $s a bad appror-matiom as has Gen found whea tMs criterionwas fm-k appliM (333) to the then-used models of Néel walls. The twovalues of this wall energy HiFeered by a,n order of magnitude at certaân flmthiclmesses, whicb showM that the models used for thts wall calculationweze verytbad approfmations, at least in that rauge of ft)m tkicHesses.

lt is thus possible to use tke dfaence, or ratio, of the energes com-puted from eqn (8.4.47) and from mn (8.3.22) ms a quntitatine measurefor the validity of the mode.l or the xxtvuzn ptions used in any micromagneticcalculation. If these nqmlwm are very diferent, this calculation is wrongand must l)e discardM. If they are reasonably close to ozmh other, the cal-culation has a good chance of being correct and xlf-consistent, and a goodapprofmation to thc real enerr minimum. It is onlya chxnce, because tMscriterion is only a necessary, not a suëcient, condition for the calculationto be corrbct. In the frst placej Brown's mqations are aISO fuKlled by themnxn-ma, aud not only the enera minima. Also, the botmdary conditionshave not been used in the above deivation of eqn (8.4.47), and a solutionof Brown's mqatbns whicb does not G,1A1 the boundary conditions is notnecessarlly an energy minlmum. However, a solution obtained by auy sortof energy minirnization is not very likely to be close to a real rrladm'um, aadat any rate, it is always better to have a nec- condition for mllvninat-ing some wrong cases tun to have no critezion at all, and so have no ideawhether the calculation Axq any meaniag at all. lt should be particularlynoted that mn (8.4.47) conusns seccd derivatives in the term witzlj VRmx,and as such is very sensitive to the devils of the maaetizadon structure,which makes this cxlterion of self-conMstzncy qnite Gectivev

TMs criterion was irst suggested (333) for a mrticular case of eqn(8-4.47) wbiG applie only to a 180* domain wall in a f1m wkich haa aun-la'dal A.nlmtropy-ln this particular case, there is neither à.u applied feldnor surface anisoàopy, and the frst two tea'ms in the frst line of eqn (8.4.47)cancel each other, so that the whole frst line of this eqqadon vankshes. Itwas then extended to a cubic symmetc 1323J, azd to a 90* domaln walljîrst in one dimension (a34) and thea in two dimensions (335). A non-zeroapplied f e1d (336) aad a moving wall (3371 weere also considered. Other self-consistency tets n.1Rn egt, which dther are not useful (295) or apply only

ENERGY YNNIZATION

T11E DYNAMIC EQUATION

to the specifc case (334) of one dimension.

8.5 The Dynnmlc EquationThe Ume-dependence of the ma&etization can be obtained directly 1omthe quantnm-mechxm-cal (axpresdoa for a precession of the maretizationin amaaetic feld, by considering the terms in the brackets of eqn (8.3.40)as an e'edtfse magnetic feld. Other methods can also be used (145) toderive the sxme result, which $s

= -,yo M x 'zftdt

where t is the time,Jzel a s.4s)% = ( .2m.c

is the rzomagnetic ratio, already mentioned in section 5.2, and g is thetLandé factor', already mentioned in section 2.1. ln some lum-, Brown'sstatic equations can be considered as a particular case of eqn (8.5.49),giving thestatic equilibrium when there is no ckaqge in time- The bouadauconditions here are the sxme as in the static =e, nxmely eqn (8.3.42) .

This equation repreents aa uadxmped precession of the magnetization,whic.ll eAn continue for ever. Eowever, actual changes of the magnetizatîonare known from experiment to decay ia a Mite time. As is the case withthe anisotropy in atioa 5.1, the dnzn phg cannot be dezived theoreticallyfrom basic principles, and is Just added as a phenomenologie Yrrn. Oneway to add it is to modif.g eqn (8.5. .48) iato

& z , 1 Aaezf = V m + Ho + H - , (8.5.48)27 Kdm

dM dM= --V;M x .l4 -p- ,-kà- dt (8.5.50)

where , is a phenomenolofcal dxmping pazameter.This form of the equation is due to Gilbert. H is actually equieent

to aa older form of Landau and Lifsbltz (145, 3381, whicN can the dmivedas follows. First M . Ls applied to vth sidc of eqn (8.5.50). The right-hand side vanishes, and therefore M - Rvldt = 0. TV result mfuzns thatdMnjdt = % from whic,h it follows that .MQ roznnsn s a con-m.nt dming themotion, and this constant can be identled with Ms1. The.a M x is appliedto Mt,II Mdes of eqn (8-5-50), ustg the general rule for a cross-product ofa coss-prodnct aad M - ?1M/d,t = 0. The resalt is

M x = -yz M xX (M x x) - vc Mszq-a-. -(8.5.51)

Substituting the right-hand side of tlzks equation for M x dM/dt in eqn(8.5.50), and reaz-ranging, lpxz!s to

182 ENERGY MTMMUATION

= -GM x N + IM x (M x m , (8.5.5 2)

where

'%';. é I

'la = 2 : 21 -L. 't' z? - R''' i' Q s

2%. p Tand h = .l -F- o'2 .42 zV2o s

(8-5-53)

Equation (8.5.52) is the older form of Lrdau and Lifshitz, which somestill prefer to use, bessdes other forms (339) which also eldst. lt can be seenfrom the derivation here that tke two forms a're mzthnmzbicdly muivalent(145, 338) if the physical constants are modifed accordinê to eqn (8.5.53).The physical interpretation ks not very dxerent either, if the damping îssmall. For a large damping there are some practical reasons (340) to preferthe Gilbert form of eqn (8.5.50).

9

THE NUCLEATION PROBLEM

It is very drë' c'ult to xlve any non-linear d''Ferential Nuation, and it is evenmore dlmcult to cllœse the appropriate physical cmse among the vadoussolutions wlzich such = equation may have. Thezefore, bdore irying anysolation of Brown's deerential equauons, it seemed desirable to ddne &stthe possiblebraach on which the required solution may be. For this purposea solution was AM sought of a set of linear diferential equations, to bedened in section 9.1, wlzich became known ms the nucleadon problem.

Thks proble.m was misunderstood by many 6om the very be#nning,an.d csven more so later, wh= Dicomaaetics became popular among thezp-rchcs of di#tal recording. Ndther the purpose nor the techniques orthe resulu of tàe nuclGtion problem seem to have been properly followed,aud there is a large number oî m017 quotations and misrepreentationsof this problem in the litarature. The older papers are oaly too often Justpresumed to say the exact opposite of what they zpxlly dé, which mayl:e due to the hct that the writers of these papers reported only certaiadetxllqj Msuming that the main csumpdons were well Hown. They wereactually known to the small number of spedalists worM-mg on them at $hetime, because practienlly evebody elsejust ignored them, but they wereunHown, and therefore misintarpreted., whea the Eeld was zevived later.

1 will try to clarià here some of the wrong czmcepts that have beenattaeed to ihe studies of nucleation. Eowever, even before defnhg whatthis problem 1s, it is worth mentionhg what it is not. The lin-xn'zxdon ofthe equations is not = approzmation, and should not be (:0zd1)* withthe physically completely HiFerent probla of the appracls f./ saimrakionia a matezial which contins poht, liney or plane imperfections. The lattercalrtnlxtions, whic,ll are outside the scope of tlzis book, are mathematicallysl'mîlnr to the nucleation problem, because they use the same linearizeddiFerential equations (although with dferent boundary conditions). TheyrtïxriMnly played aa important and a crudal role (145, 268) ia the initialdcvelopment of miœomagnetics in general. But they have nothing to dowith txe nucleadon discussed in this chapter.

9.1 Doënltion .

Let a ferromagndic body be Erst put ia a magnetic îeld wkicà is largeexough to saturate it in this feld's dizecfon. Let this îeld be then reducedslowly, to avoid dynamic efects. If necessary, the feld is decreased to zero,

184

VTHE NUCLEATION PROBLBM

(a) .=(b) >(c)

FIG. 9.1. Two mechanical aaaloguo of the nucleadon probiem-

then increased slowly in the oppodte direction. At some stage during thisprocess, the state of saturadon along the original direction of the appliedGeld must stop to be stable, and some change must start to take place,because tafte.r a1l) the samplc must eventually be saturate in the oppositedirection. The feld at which the orighal saturated state become unstable,and azkv sort of a chauge in tEe magnethadon conîgmration ca,n tnst glcztis called the ncleation flelï The name is somewhat misleading, because(341) 'nucleationl seems to imply that something happens at a particularpoint, around a certain n'uelenn, wher= in the present context, this termis used for something which may happen all over the crystal. Brown arg'uedall Ms life against ths use of the word, but he had nothing better to ofemaad the name stuck and was meed by everybody, including Brown himself.The important point is that this nudeation haa an uambiguous meaning,which is az defned by the foregoing procv. lt must be emphzmlzM that thedefnition contals the history of the applied îeld, which must be the c%ein any defmition involving the hrsteresis whicb Ls part of fenomagnetism-

ne coacept ok nudeation feld is analogous to the critical force whkhis ne- to bend a beam in the Gperiment shown schematically on theright-hand side of Fîg. 9.1, whîc,h îs partly hoed on (341j- If an mlxxtfa-cbeam is pushed from b0th sides by a force (reproented by the arrows), asshown in (a), nothing happens at ârst-With an ïncreadrlg force, a 'criticapvalneis r-Mbed, at whicb the beam suddenly buckles to a particular shape,whicb is an eigenflmction of a (Main dxerential equation, as showa in (b).H the analogous case of a ferromagnetic crystal nothing happens when thefeld is Grst 'lnduced till a certain value of the feld - the nudeaiion feld- is reached, when the magnethation suddealy lbucldes', or Ganges inanother way; and this czaage is alK aa eigenfunction of certaân deerentlnlequations, as will 1>e seen in the following. It must be particulazly empha-sized that during this 6rst stage, when notking happensj there may well bestates whose energy is looer than that of the satnrated state, but they azenot accessible to the system. This pœssibility has already been eacountered

DEFINITION

in the case of the Stoner-Wohlfarth model in section 5.4, and is furtherillustrated by the second mechanical analogue on the left-hand side of Fig.9.1. In the beginning the saturated magnetizadon state is the only energyxminimum, and is amalogous to the ûttle ball phced at a single depressionof a sm00th surface When the feld passes zero, the saturated state inthe opposqte direction already has a lower minimum, and the situation isanalogous to the ball placed on the surface shown in (a): there is alreadya lower pit, but the ball cannot roll there, because of the energy barrier inbetween. This ball cmn roll down only when the surface ks further distortedinto removing the barrier, as in (b), which is analogous to the nucleationfor magnetization reversal to start. Therefore, just choosing a particularmagnetization as a function of space, and proving that its energy is lowerthaa that of the saturated state (or any other statel for that matter) iscompletely meaasngless, and does not prove that the magnetization willactually choose that state. It must be shown that the lower-enera state isaccessible to the system, and that the situation is not as in Fig. 9-1 (a).

It should also be noted that the surface on which the ball is placed inFig. 9.1 is actually a representation of a multi-dimensional surhce in thefunction space, and in the actual case of magnetization reversal there aremany possible paths for this ball. Therefore, showing that a particular pathis blocked by a barrier does not mean anytidng either) because there maybe a way around it. ln principle, C!J possible functions must be considered.

For the elastic beam shown in Fig. 9.1, all the harmonics of a given solu-tion are usually also solutions of the diferential equation with its boundaryconditions. An example is shown schematically in (c), and there is generallya whole set of such solutions. ln the caseof this beam, alI these higher-ordersolutions have larger eigenmlues, namely need a larger appëed force, thanthe basic solution of Fig. 9.1 (b). In this case, none of the other solutionshave any physical meaning, as can be seen fz'om the following argument.Suppose it takes a certain force Fz for the beam to buclde as in (b), andsuppose that ii theoretically taka a force Tk > Fz to create the deviationof (c). In order to apply the force F:, it ks necessa't'y to pass through theapplication of the force Fz , at which time the beam already changes intothe confguration (b) . By the time F2 is rcached, the initial conditioas (a)do not efst any more, and a theoretical transition from (a) to (c) at theforce Fa cannot be realized. The higher-order solutions may apply in.oà/zerc'ases, e.g. if the beam is tkampe.d at its centre in such a way that (b) cannottake place, or if the hrger force is applied very fast, when dynamic con-ditions may prevent the formation of (b). For the nucleation problem asformulated here, the higher harmonics have no physical meaning, beiausethey cannot be achieved. By the same token, in the case of magnetizationonly the lcrges't nucleation feld has a physical meaning. lf something startsto reverse at a feld Hz, the reversal will continue on that path, and anothernucleation which can theoretically take place at H, < Rh does not have

186

the Mtial conditions of a saturated state any more, an.d hms no pkysicalmeaning. Foz any case of nudeadon it is thus necessazy to fnd only tkehrgest posible dgenvalue of tke appropziate dxerential equations.

The nucleation process, a: a stact of the reversak is studied by Ifsead-zing Brown's equadons dened in section 8.3, namely by leeaving out df tkeseequations tke higker-thaa-linear powers of tke maRetîzation componentsia the dizections perpendicular to tke applied Geld. TMs Bnezm-zation isn/t = appremation. Jt is only a manifœtation of the requlement ofcxmtinuïfr. Every chaage of the magneeadon structure must siart witk asmall ckaage. Thereforey everything is bnear in tke b - '

g. The idea isthat onœ tke correct eigenfnncdon for the nucleation is known, it is goingto be possible to study t/e zest of tke process by soleg tke non-linea.requations for tke cmse which starts with this particular eigenfnnction, andnot movein tke anxrk tllmugk all sorl of mathematically possible solutionswhlch have no physical meanlng.

Tke lixpxrlxtion of Brown's equations <11 not be done hcre for themost general case. Jnstead, tke fozowing restzictive assumptions will frstbe made.

1. 'Phe appliu mavetic feld, H., Ls homogeneous, aûd is paraltel toan easy n='s of either cubic or nm'a='x1 aaisotropy. Tkis mssumptionwxs made in clî studies of nucleadon publsshed so far. In peciple,a theory could nlM be deuoped for other magnetic 6.e1ds, but it hasnever been tzied- Tke xcond part of assumption 1 follows, because ite= be shown tkat it is imp-ible to reMh saturation in any fnite,komogeneous feld in a dlection whicx is not aa easy Jt='K. If z Lsckosen as the dirlctkon of the magnetic feld, it is readily sœn thatto a fus't order in ma an.d mx,

- + = -zA%ru, (9.1.1)0- zp,: 0mz

THE NUCLEAXON PROBLEM

where m staads for either 'u?u of mn (5.1.7) or 'lz?o of eqn (5.1.8)-In eitker casea Kg is noé neglededj as it haz been in some othercalculations ia tMs book. In spite of the presentauon ia (342), tkea'eis just no Grst-order term in tàe expresions witk Kz, and it doa notenter the rzzaczetïtm problem. A similaz expression applies for mv-

2. The sample is aa ellipsoid, and tke feld Ls applied paralld to oneof its major axes. The &st part of assumption 2 is hevitable if akomogeneous feld is assumed, because only inside an elltpsoid is thedemagnethsng Geld homogeneous, sa section 6.1.3- The second partis not essentlnl, an.d ks only mtroduced here for the sake of m'mplîcity., .

Some calculations have been reporte (343, 3M, 34,5, 346j for a feldapplied at an aagle to a major ikx5K, but onlyfor a Tez'.y' limite numberof casey wMch wiz be iaored kere. According to section 6.1.3, the

DEFNTION l87

demagnetizing f e1d at the saturate state, before nucleation, is aconstant, aad is paratlel to z. Txezdore, the toti îeld of eqn (8.3.34)which ka.s to be used in Brownhs equations is

OU (??JVx = - - . & = - - ,01 ' = %where Nz is the demagnething factor i'a the z-dizection, in the sat-'lrxlzx,l state bdore nndeationj aad U is the potential due to thestarting dekiat,ion from saturationy and is of the order of vzz an.d mv.

3. The matedal is homogeneoust and %M no KnrfMe aaisotropy. Only afcsw cmses with a non-zero surface Ysotropy (146, 347, 348), or withcertain inhomogeneitie (34% 350, 351, 352), were ever s'tudied, aadb0th *1 be ignore here. Only some of their implications will bebrieây disrmnqtvl in Gapters 10 and 11. As mentioned at the end ofsœtion 8.3, the bolmdarz conditions are in this case

OUHz = Hu- NaM, - , (9.1.2). az

(9.1.3)on the surface. )

Using a11 thue assumptions in eqns (8.3.37) and (8.3.38), and oYttingall terms wàic,h are higher than liaear in mz or mj or U, includfnp a termS'UCX aa mzu, which is also second order when Lr zs of the order of mz ormu, yields the dlFerentii muations

0mt 0mv= = 0

Dn >

Ah .-

((7V2 - 2r: - J'pfs (k-To - Nzasfalj mx = ât'a - - m

os (9.1.4)

and- c alhnI(7V - 2Jf1 - Ma (Fa - Nz x%fs )j my = -%Ix oy (9.1.5)

inside the ferromaretic body- MM of the earlicr studic started with aaextra term of the form gïgmzmw in the anisotropy enerr daits whlc,itadded aaother term to 0c.11 of these KuaKons. But then this term wastaken to be zero at a later stage, and at any rate it is not usually part ofany anisotropy enera.

These equadons have to be solved together with eqn (9-1.3) ms thebotmdyy conditions, aad simttltaneously with the ecuations and boundaocondittons, (6-1-4) to (6.1.6) wlzich defnethe potential U. For the lin/xrlmedcee of - only Xst-ordervrmK in mx and niy, the diferential equationsare

0mz 0mv aVzr7u = 4xMs + , V Zut l:= 0, (9.1.6)0z oywith the boundary conditions

l8S

(9-1-7)

on the sudace, as we.ll as the regulvity of tke potential at infnity. AII theseeqnations have to be solved for all poaible eigeneues of the applied ield,S'a, aad then thelargest of tkem has to be chosen- As has bam explxined inthe foregoing, only the larges't allowed value for Hu has a physical meaning.

THE NUCLEATION PROBLEM

&Uu OU tOin = U'out , o ' - ---*X = 4*.Km ' zl ,n, on

9.2 Two EigenmodesBron wroie this set of linearized equations (353, 3541 in 1940 for thesake of aaother problc. Only 17 years later did he formulate them as thenucleation problem (355) and realize that two analydc solutions could bewritten right away) b0th for a spkere and for an -mGnita cirfmlnr cyliader. Itactually turaed out that one of them could be generstlt'zed to any ellipsoid,and the other could be genernll-qed to an ellipsoid of revolutkon, namely onewkick has two equal axes. Tkese two modes will be dermibed here Erst, inihis more general form, Vfore addrving the problem of other pfwwlbledgenfundions, whick Brown (3554 presented at the tMe as a gap in thetheory. As has already been Irentioned in the previous =tiop only thelargest eigenvalue has a physical meeng, so that any one eigenfunchondoes not mpnm ver.y much.

S.2.1 Coberent Jbtafi/nJT b0th 'rzzz and my arc constats, eqn (9-1.3) is Gàlfmed. The volumeeargeis zero, and eqn (9.1.6) for the potential becomes V2U = % b0th insideand outside. Jt thus reduces to the problem of a komogeneously magnetizadelpsoid discussed in section 6.1.3, which leads to a homogeaeous âeldinside, with

Pon r PLV r= Lsz-vvmv and = wNvMxzr7v, (9-2.8)t'iz ou

where N. aad Nv are the appropriate demagnething factors. Equations(9.1.4) and (9.1.5) A.rm in tkis case

-2.R-1 Q.R'I- + Hx + (N. - Nz)Ms m. =

u. - + Sa + LNv - Nz)Ms mv = 0.Ms i s

(9.2-9)These equations aze all tkat is left in tMs case from the whole set of thelizm-m'Rz.vl equations, and nudeation cmn take place once they are ftzlMed.

To r=ind the reader of what tMs Gcnlxtion is all about, the stari wmsa saturated state m. = mv = 0 at a large positive H=, which was laterreducM throtlgh 0, and started inceasing in the opposite direction. Thenudeation a a magneiqation reversal becomes possible wken Hx rpmrhesa value that allows a (small) deviation 1om the saturated state. lt ea,n

TWO EIGENMODBS 189

happen when these equadons can be fxllilled /or ihe Srst àfrnz with eithermz # % or 'rr/v # 0. Therdore) this nucleation is aiieved wken the appliedfeld Na zeackes a Yuethat makes one of the square brnrkets of eqn (9.2.9)

x .pass through zero. lf the ellipsoid hms a symmetry for revolvhg around z,whick means that Nu = Ny, the eigenvalue is the mnme for rotation inthe m- or in the p/-diretion. If they are not the sxme, the rotation will betowards the lonner =-!R, because it has a larger (less negative) eigenvaluethazt a rotation in the other diredion. Suppose the Ionge,r aMs is z, namelyN. < Nuk then the nucleation is for my = 0 and txltoq place when H. .

reaches the nudection éel: value of

2A'zHn = - + (Nz - Nz) Ms.M. (9.2.10)

ln this mode the mavedzation rotates in the sxrn e angle evermherethrough the Gpsoid, and it is therefore Hown a.s the coherent rotationmode. The nxme Srotation in nnlsonl was alnn tried (356j for a while, but itdid not catch on. Actually, it is Just the Stoner-Woblfn.rk% model, studiedin section 5-4 for the more general caqe of a feld applied at an angle to theeas.y xvkq- In $he pxesent cmse the seld id parallel to the easy ads, wMch is0 = 0 in the notation of that section. In section 5.4 it was assumed thatthere was only a crystalline xnl'sotropy, with no shape anksokopy, whichessentially means a sphere. For that ca'le it was seen that at zero =gle,nothing happens till the feld renzthes the value of the frst term in eqn(9-2.10) here- Th% in sedion 6.1.3, the second term of the nucleation feldwas introduced for the case of an elllpsoid without Misotropy, namely with.JQ = 0. Herey mn (9.2.10) is for the comblation of both esotropy terms,but ouly for the cmsewhen the easy axes of b0th ate plkenllel to the appliedfeld. H this case, their 'values are just added togeiher, at l/uAt dming thestart of the deviation 1om saturation.

The Stoner-Wohlfxeh model, which started as a model namely aa apœtulated structm'e of t:e mavetization in space, has thus been skown tobe a mode, unmely an eigenfnnction of Brown's equatio=. As such, it is arealenergy msnimum, and not just aa arbitrary confgaration for comparingener#es as in the domain theory, whiG Brown tried to avoid (see sedion8.3)- It is not the end of the road yet, becaux lhlK mode will actnally beused by the physical system only in cues in whick it is the mode with t:elpxqt negative uutleation Neld. In order to fnd out if it is, all the othermodes must be investgated, aad compared with this mode.

9.2.2 Magnetizztion Jurlfn.qAnother mode wlzic,k Brown (355) fouxd to be a soluhon of the liaeltriv>dxt of equations applia only to an ellipsoid of revolution, or at l/mAt nobodyhas tzied to generalize it to any other ellipsoid. ln cylindrical coordinatap, z and 4, what has become Hown ms th'e curling mode is the solutîon of

190

Brown's equations for whiciz

m. = -Fçp,z)6n$, TP.P = F(#.,# cos/, Qa = D-out = 0. (9.2.11)

'rEE NUCLEATION PROBLEM

It is an arbitrary set of constrlu'n? on the soludoa, whose onlyjustifcationis that it nwrka, namely that such a solution does ezst for an.y ellipsoid ofrevolution.. 1.n ozder to see that it does, it is obviously stlldent to substitutethese constrints in eqns (9.1.3) to (9.1-7) and see that thm.e is a solutionto the constrained set. Substituting thus eqn (9.2.11) ia eqns (9.1.4) and(9-1.5), it is sen that they are Wth fnlfllled if

- Ms (Hu - NzMsà F(p, z) = 0 .

(9.2.12)Equakon (9.2.6) is obviouly fnl4lled, and so is eqn (9.1.7% even thongh thelattex Alkkes a little thinking about to see that eqn (9-2.11) actuuy leadsto m . n = O on the surfre of any ellipsoid of revolution. The poht is thataccording to this deGm-tion of F, tkis F is actuvy the component of m inthe direction of the coordinate $, namdy m4. Tkis commnent is parallelto the surlce i.n any body which has a cylindzical smmetrsin particularan ellipsoid of revolution. Therefozey the only muation which is still leftfrom the ori#nal xt is eqn (9.1.3). Substitution in it yields the boundarycondition

0F= 0, (9.2.13)

az +

-! ta -

1 +

.,?z

; - zs,qclv ,s v v

on the suAce, where n is the normal to the surhœ.The assumption qf mn (9.2.11) has thus reduced the three-dimensional

problem to a tweimensional one, wldc.h is not diEcult to solve in theellipsoidal coordsnate system. However, since tkese coor&ates may be tooadv=ced for some readers, the solution will be expressed drst for the twocmses of a spkere and an t-nAnite circular c'yBnder, ori#nally studied, almostsimult=eously, botà by Brown (355) aad by Ftmi e'l cl (356J. There is,however, a big diference between the two which has ben forgotten duringthe years, and whiG is worth emphaaizing.

Brown started from the dilerential equations, and guessed a pmicularKlution for a sphee aad for aa in6nl-te eylinder. Frei 6i al started from aparticula.r functîonal lrm for the magnetization, and compared its enea'awith Gat of some other mpdëls. In their paper, they presente tlhe curlingmode as an arbitrary model for a sphere and for an in6nite Vlinder, becausethey did not How iat it wms the same soludon of Brcwn's equationqpublished by Brown a few months eearner. Since Brown did not give a nameto this mode, au.d since people nsually feel that when something is givena name they understand it bettœ, the paper of n'e.i et cl (356:, in whic.hthe name 'curling' was fzrst ixvemted, became much bdter Hown and citu

TWO EIGENMODES 19l

than the paper of Brown (3551. In his talks and publications, Brown keptemphasizing the strange coinddence of the same solution being worked outsimitaaeously and independently in two places, but he never mentioned'this diference between a reversal mode that obeys his equations, and amere model, probably because it was so obvious to Mm, and to everybodyelse at the time. The unintentional result was that too many people werelef4 with tàe impression that (3555 was just the same ms (3561) and was notworth reading. Thus, too many books aad reviews give a schematic pictureof what the curling (looks like'; usually only in a.n infnite cylinder, butnot the mathematical defnition of the function, as #ven here. And papershave been, and still are, published with all sorts of models for magnetizationreversal in whic,h they compare theirs with the tcurling model', e.g. E35'C. Itis impossible to convince them that the curling is not a modei, aad cp.nnotbe treated on the same level as their arbitracy models. lt is a nucleationreversal mode, which is a solution of Brown's equations, and as such cxnonly be compared with other reversal modes such as coherent rotation orthe other modes discussed in section 9.4.

9.2.2.1 Infnite Cylindcwr For an infnite cyDnder the norrnal n to thesurhce is parallel to the coordinate p, and it is only necessary to considerF which does not depend on z. Eqn (9.2.12) is the well-knoWn diierentialequation for the Bessel functions. It has two solutions, one of which is notregula,r at p = 0, and cannot be used ms a solution. The other one is

F G Jzlkp), (9.2.14)wldch is a solution of eqn (9.2-12) provided that

Ck1 + 2.&% + MsHu = Oj (9.2.15)

where tbe term with Nz has been omitted, because Nz = 0 for an infnitec'ylinder: see section 6.1.3. The boundary condition (9.2.13) is also fnlflled,

dh @p) = 0, (9.2.16)dp zzap

where R is the radius of the cylinder.Equation (9.2.16) has an l'446115 te number of solutions, out of which only

the smalLest one has to be considered, because the larger ones lead to amore negative nucleation feld, which cxn never take place if a less negativeone efsts. Let çz be the smallest root of

dJz (ç)= 0 (9-2.:7)dq

l92

(which Ls t21 = 1.8412)- Then the nucleation feld for titis mode is, accorrlingtc eqn (9.2.15),

2A% (k21H.v = - - -z . (9.2.18)Mk R MsCompariug with eqn (9.2.10), and tazng into account tàat for an l'nfna-tec'ylinde,r N= = 2* and Nz = 0, it is *e,11 that iJ tbere i: no ofher mode, mag-nedzation reve-l ill a.a '-n*nite cylinder should start by coherent rotationif JL < &, aud by curliug if R > J?.o wkere

THE NUOLEATION PROBLKM

/''t('Ff,ll , g g .jg;Bm = h/ / -2 x ' ( ' -Ms

because it is always tke largat Jo whic,h counts: ms explained in xction9.1. This svtement is not true if there ks a third mode with a still largereigeneue for Sa. This possibzity will be further discussed in sedion 9.4.R'e,i ei c! (356) iatroduced tbe redlmed rcdi'tla,

R I Gs = -, with z?o = l j ,a j/ 2M, (9.2.20)

and this notation was later used in many papers on micromagnetics. ln t/isnovtion, the tumover fzom coherent.rotation to czzrliug is at the reducedradius

R ez 1.8412Se = -X = ss 7e, 1.039. (9.2.21)

9.2.2.2 Sphsre The second case considered by Brown (355) and by Freiei al (356) was that of a sphere. For this case, the cylindzical coozdmatesp and z are changed to the spherkal ccordinata r and 0j with 4 kept thesame. In these coordhates, eqn (9.2.12) transforms into

g :2 2 t'i 1 02 cos 0 ê 1c + - + ;s + a ya o y

-

p--r-w.s mk 0r2 r'W 0e r s

4r-2Aï - Ms S'a - 'y'Ma F(m, #) = 0 v (9.2.22)

because N> = 4*/3 for a sphere. One of the solutioms of this equation is

F (x j1(k'r) giu.#, (9.2.23)where Jz is the spherical Bessel function, which can n.1M be exprased interms of the trigonometric functions,

sin z cos n.i @) = oa - . (9-2-24)X

TWO BIGENMODES

It is seen to be a soludozb provided tkat

(9.2.25)

Actually, Brown (355) also comddered other solutions of the same equation,but they had a smallc (î.c. more negative) nudeation îeld than the one illeqn (9.2.25), and ms s'uc'll a2e of no interest- The whole eigenmlue spectrumis left to be discussed izt sedion 9.4.

The boundary condition (9.2.13) ks fttldlled if

djj. (kr) =r 0, (9.2.26)dr uar

193

4xc:2 + 2A% + Ms Na - Ms = 0.Y

where R here is the radius of the sphere. Tbis equation has a,zl ln6nt'te aum-ber of solutions, out of which only the malleat one has to be considered.Let qz be the smcilest root of '

4'/1(c). = () (g.2.27)*

(which is q, ;4$ 2.0816). Thea the nucleation Eeld for thks mode 1, Rcordixgto eqn (9.2.25):

2A% Cd 41Hn = - . - - - + yMs. (9.2.28)Ms AzMsCompxring with eqn (9.2.10), and txMng into accout that Nn = Nx fora sphere, it k seen that f there is m) other mtlde, magnetizatâon reverxia a sphere should start by coherent rotation if R < &, aad by cuz'ling if.R > J?,c, where

h 3C& = , (9.2.29)Ms 47

wid.c,h is rather similar to the exwession for aa infnite c'ylinder. H thenotation of Fre,i et c1 (356J, the tllrnover from cohezent rotatiot to turlingin a sphere is at tke reduced radius

(9.2.30)

9.2.2.3 Bllipsoid 0/ Revolnkion 80th early studies of curling g355, 356)speculated that this ruult cou)d be utendu to a prolate spheroid: fœwlkic,h the curDng nucleatîon feld should be

'a rq Cq2fzk = - -

as.i + NzMx, (9.2.31)Ms R s

R 3 3Sc = --1 = qz - = 2.0816 - = 1.438.& 2r 2/

194

wkere R ks the snmi-axis of the ellîpsoid in a direction perpendicular to t:ef e1d directson z, and q is a parame/r whose value is between qï and qz.By solving eqns (9.2.12) and (9.2.13) in terms of the ellipsoidal harmbniœ,it was latez shown that equ (9.2.31) is indeed the nucleadon âeld for thismode,

'

both for a prolate aad for aa oblate spheroid. The Nrameter q is ageometrical factor, which depends only on the aspect ratio, zp,, namely theratio of the ellipsoidal axK, and does not depend on the properdœ of thema/rial. 1ts value is a monotonic-ally decrexm-ng function of m, whicà variesfor a prola,tf spheroid between the limits of qn for the sphere with m = 1,aad qz for the A-nGm-te cylinder, wit,h m = x . It used to be taken 1om anold, and not 'vez'y aœuratey plot of q r,s. m, but now ît can be obtained (358)fzom the follcwing polynomsxl, whic,h is correct to îve si/zifrltn t digits,

THE NUCLEAXON PROBLBM

q = 1.M120 + .48694/m - .11381/-2 - .50149/m.3 + .54072/+4 - .l72/mS.(9.2.32)

For aa oblate sphezoid, q keps increasing monotonically with decemsingr?z (359), fzom qg for the spherey to the value qs = 2.115 in the Bmt't of aninlnite plate with rrz -+ 0. The càange in this whole reson isj thus, verysmall, and a eozustant value a q ra 2-1 Cs corred to within 1% foraay oblatespheroid. Or, ?/x ra 1.4 ma.y be n!uvl (3591, which is correct to within 2%.

Compaz'ing eqn (9-2.31) with eqn (9.2.10): it is seen ihat for aay eLpsoidof revolution, nucleation must be by coherent rotadon for =a11 rxzll-i and)by cuzling for larger onu) and the Ccriticap radius for -

g betweenthue two modes is

(9.2.33)

pzovided that there ks no tMrd mode wh- nuclGtion Neld is large.The curMg mode is zmt really lsmited to c,ax of drfmlar symmdry.

For the caae of a prism, whic,h îs infnite i.a the zzirection, but has arectangnlar cross-section i.a the zp-plane, the.curling is ddned (359) as thereversal mode for whlc.h mx is an even ftwction in tr and an odd function iny, aad. rzv is aa evea Nnction in y aad an odd ftmcuon in z. The potentialfor tMs mode is not zero, and neither is the magnetostatic enerar, but itAlKn yields a nucleation feld which has a term with an essentially 1/R2zependence-

9.3 Tmeo&* te SlabIt has been shown in section 9.l that the only solution of Brown's linexrszedmuations which %nA aay physici molkn-tng is the one which has the largœt . -

nucleadon âeld. The two modes described in the previous section do notme= very mucà nnlœs the whole dgenxmlue spectrum is analysed, and itis shown that all the other pœsible modes have a smalle,r eigenvalue- The

r--c 5--2q /Rc = -q -c , or Se = q = ,Ms A z hz

INFNTE SLAB l95

6mt of suc,h studie of the whole spectrum was for the case of an ivnitec'ylinder. It showed (360) the existence of a thrd mode, but eliminate a,llpoebilities of a fourth one- This c%e is atypical, and its resulî are not

' as condudve as other cces, which v,ill be discussed in the next section.1ts algebra ks also rather complicated for a start- Therefore, tNe pAdpleof coverhg the whole eigenvalue spectrum will be dnmonstrated Nere by adetaâled study of an infnite plane, ms presented in (3614. This c%e is alsoatypical, and its xsults are zather ambiguous and of no particular interœt(362), as usually.happens whenever an ''n6nity is mssumed îa maretostaticproblems. However, the zrletàW is the same as in the more complicatedrxdaax, and it is pxqier to undeataad this method by consideriag this simplerxqo fm$.

Therefore, consider a plate whic,h is iaâaitf in both the z- and the y-irections, and Mends over the flnz'te raage -: S z S c in the directionof the applied âeli In an inf nite material, any non-divergent Gtnction ofspace must be a periodic functioa. With an appropziate coasideration forthe symmetry propertie of etms (9.1.4)-(9.1.6), this pcriodicity meaas thatthe most general solution rztn be written in the form

mz = A(z) SA:'Z - zc) eostnp - &ô), (9.3.34)mu = S(z) coslàz - *o) sintzzv - #c), (9.3.35)U = 'utzlcostàz - zn) costnv - #0), (9.3-36)

where k and n are real numbers, and .4, B and e, are functions whiex haveto lx determined- Substituting i.a eqns (9.1.4)-(9.1.6), and noting thatNz = 4* for an snflnite plate, Wan- Nz = Nv = 0,

c ( P - :.2 - .42 - 2A-z - Ms tSa - 4xK) A(z) = -kMs'uis(z),02

(9.3.37)O a 2C c

,- k - s - 2.&% - plk (Nw - 4xMs) Blzj = -sMa>a(z),dz(9.3.38)

tfz z .

dzg - P - n 'tqatz) = 4xMs (kz1(a) - sS(z)) , (9.3.39)

Jz -

- kl - .p.2 ,tt t(z) = 0. (9.3.40)dz2 1 eu

The boundary conditions are obtnln ed by suMtitutin.g the sxrne equatioas,(9.3.34)-(9.3.36), in eqns (9.1.3) and (9-1-7)- The 6m-+ two conditions are

(IA dB- = - = 0.dz sc dz a=+.A=

(9.3.41)

196 THE NUCLEATION PROBLEM

The other boundary conditions are easier to incorporate if it is noted frstthat eqn (9.3.40) has only two possible solutions, one of whic,ll divergc atinfnity. I1s only solution whicà is realar at '-ninit.g ks

= vse k.+o-(ezp) (g.z.4a;cout ,

where the upper sign applies to the reon z > c and the lowe sign appliœto the re#on z < -G and whe.re F+ and 7- are two integration constaats.Substitutin.g thks solution în the ret of the boundazy conditionsl they areseen to be

dwn a a i,. (asgsgglw'uzlicl = Fc!z , and = ry k + n :i: ,dz mukc7

which completc the reduction of the ImJIM dœeratial equauons to a setof or ' ones, i.a one dimension.

The cohexent rotation mode is the particular cmse k = p, = vzin = 'tlout =

0, with either W = 0 and B = const, or B = 0 and .4 = const, accordingto the defnition i.n section 9.2.1. It rAn be verKed by su%titution thatthis cmse Ls hdeed a solution of a2 the foregoiug equations, and that thenudeation feld is the same Br rotation in the A-direction or in the :>direcdon, and îs

2fGH= = - - + 4rMsj (9.3.44)Mswhich is a particùlar ca% of eqn (9.2.10), for Nz = 0 azkd Nv = 4çr. Butthis eigenvalue is degeaerate not only with respect to the direction of therotation. If the infmite slab is taken as the lsnn-t of a,n obla* spheroid forwikich R -+ x, eqn (9.2.31) for the nucleation by cnrlin,g nlnn tends tothe same value ms in eqn (9.3.44). The point is that an infnity is neverwemdeved h magnetostatic problems, and it must be specifed the liml'tof whîch shape this infnity is. ln a 6nlte body, a coherent rotauon involvedoing work aYnst the magnetostatic for= due to the smface nhn.rge onthe <de towards which the magnetization rotates, but does not ilwolveany work agaimst evfhzmge, because all the spins a:e aligaed parallel toeach other. On the other hand, the curliug mode do% work agalnst ex-change forces, because there is a spatial delmdence of tke magnethation,but doe not involve any work against magnetostatîc forces because thereLs neither sorfnzw nor volnme charge. In this atypical case of an infniteslab, the excAange contributlon to the curling wnisha because the radiusin ininlte, while the magnetostatic contributkon to the coherent rotationvaaishes because there is no smozw in f/he directton of rotation when theplate eex'tends to infnity in the zpplMe. ln this case, the only barrier isdue to the anisotropy energy, which is the scm.e for 50th mode, aad theeigeneue tums out to contain only the A'1 term--Howevem it is obdous

INFINITI SLAB

that the vanishing of oneterm is not the same ms that of the other, and thereal physical limit depends on the way of approach to this infnits as oftenhappens in many problems in magnetîsm. 1.xz spite of all the azguments in(3611, the conclusion that only coherent rotation takes place in this plateis only due to the separation of 'variables in Cartesian coordinates, whichimplies approaching the infnity ms the limit of a growing square plate. Ifit is approached a,s the limit of a growing oblate spheroid, the mode at .inûnity is the curling mode.

Before proceeding, it should also be mentioned that the argument aboutthe periodidty is not strictly corrcct, although it has been used i.n otherstudies, in particular for the cmse (360) of a.'!l infnite circular cylinder. Inprindple it ks possible to imagine some sort of a loczzlïze mode, which doesnot spread all over the slab, aad such a mode need not be periodic. It isnot possîble to build such a mode by the separation ol vadables technique,as used in writing eqns (9.3.34)-(9.3.36), but this shortcoming does notnecessarily rule oat the possible efstence of a locnlszed mode. This problemwill be further discussed in the next section. Here it is suldent to say thatthe inadequacy of the separation variable into a function of z times afunction of y, etc., is another manifestation of the infnite dâmension ofthe sample. The problem i.s ndt encountered in any Enite ellipsoid) forwitich a11 the possible modes can be written ms a series in the spheroidalwave function, and it is not necessary to supezimpose any extra assumptionwhich is equivalent to the present assumption of periodidty. If the infnityis approached as an appzopriate limit of a fnite particle, the results a'rebetter deûned than they aze when the start is 1om a particle which isinfnste in one or more dimensions. As has been mentioned already, there isno meaaing to infnity in magnetism, and it is always necessa'ry to speciàin which way this infnity is approached.

laeaving this problem of inûnity for the meantime, and accepting eqns(9.3.37)-(9.3.39) ms the most general caase, such a set of three second-orderdiferential equations should have a solution with six arbitrary integrationconstaats. Therefore, any solutjon which has such six constarts is the mostgeneral one. I.a particular, if it is shovn to be a solution, it is sulcient totake a solution of the form

6

A(z) = V Az'elLsn ,f=1

6

S(z) = SçeJ<ri=1

6

'fzutzl = ULela' , (9-3-45):=1

where /.sà aresix complex numbersy provided that six out of the 18 constantsA'l B6, aad Ui are jndependent of the others. Substituting eqn (9.3-45) inthe diferevtial equations (9.3.37)-(9.3.39), it is seen that the conditions forit being a solution a're

l98

(C Lybx - :2 - 2) - 2A% - M: (JQ - 4*Ms)j m + lMsw = 0, (9.3.47)(p1 - P - n2) w = 4xMs (kA - nBè . (9.3.48)

For ear.lt value of 1 K f K 6 these are tkœ homogeneous equationsi.a A, m and vo and Ge condition for them to have a non-zero solutionis that the determinant of the coecients vauishes. This detonnina.n t is athird-order polynomial in pl , and aa sucx shodd llave six (complex) rootsfor ;ti. For each of these roots, eqns (9.3.46)-(9.3.43) eAn be used to solvefor two out of the tlzree constants .&, Sf aad kz: in terms of the th-lrdone, thus leaeg sîx arbitrary intevation constants, widch mezas that eqa(9.3.45) is the mœt general soludon, aud contains all po%ible mode. These

'

sLx constants, with the t'wo additional oaes 'k%, should now be eœuatedby the rmuirement of fulflling the eilt muations for the boundary condi-tions in eqns (9.3.41) and (9.3.43). There are dght homogenecms equationsfor determînlng these eight constants, and the condition for a non-zero r,o-lution is thltt the de#orlm-nxnt of the coeEdents vaaishes. Equating thisdetprngnn.n ï to zero then yields the allowed values for the applied feld Hx,aad these are the eigeneua of the problem.

Such an akebra is not triviak. but it is stMghdorward in pedple.Moreover, the present cmse of an ln4nlte plate is particularly simple in thatit can all be cvzied out analytkally. The tàird-order determinant can befactorie- (361) into a quadratic aud a linear equation in Jz?: , and a11 sixroots rltn be mltten in a closed fozm. The result Ls (361q that all otàermcdes have a more negative nudeation feld thaa the coherent rotatioamode) and as such can be ignored as being physicldly unatteable.

The details of thae other solutioas wlll not be given here, because it issimpler to employ a technique (362q wlzicN has nlm proved a vea'z mwerful*ol h othe.x cxses. The method Ls lomed on calculating aa uppe-r boundto the nudeation feld of a certe mode) namely a value which is provedto be lvgex than or eq'aal to the t=e nucleation âeld of that mode. Ifth2 upper bound is found to be smaller (ï.& more negakive) thau that ofanother mode, the actual nucleation âeld of the Erst mode twhicil is nothrger than its uppe.r bound) is certxinly smaller than that of the secondmode. And siaœ only the largest nucleation fe-ld hnq a physical meaaing,any mode which Ls shown to have a smGer nucleation Neld than that of

. aaother mode ks of no iaterest, and may be safely left out. Even when twomoda hàve the same nucleation seld one of the.m may usually be left out.

One way to calculate an upper boud is to drop a vsiiine ezte-rgy lzvm,

thus decremshg the energy barrier, aad meng the reversk pxm'e thau itrpally is. In the present cmse it is czmvenient to obtain an upper boand to aset of modœ l)y dropphg the mMnetostatic enezgy termj which is knownvto be a non-negahve term tsee section 7.3.2). lt ks clear fzom the derivationofBrown's equations in secdon 8.3 that droppiag the mxgnetostaiic enerrte= is equivalent to writing 'ttuz = 0 on the visbt-hand side of eqns (9.3.37)

T% NUCLEATION PROBLEM

INFNTB SLAB 199

and (9.3.38), aad ignoriag eqns (9.3.39) and (9.3.43), befm the subsKtu-tkon of eqn (9.3.45). The dllerential muations for A(# and #(a) nm thenideadcal, and either one of them may be used fœ the upper bound. Fozlmmple, w1t,h 4(a) = 0 t'he most general solutioa Ls

BLz) = Bzegz + fac-e (9-3-49)1

with the two arbitrary constants Bï and B,. It is a xlution provided that

G' (p2 - k2 - 'rz2) - zft'y - Aqzf, LHu - 4rrMsl = 0. (9.3-50)Substituting eqn (9.3.49) in the boundazy conditîons, eqn (9.3-41), yieltks

#$ (Jhep'c - sae-mc) = s (.s,c-- - ac/'c) = c. (9.3.51)These two equaKons have a common non-zero sohtion if and oaly if thedet>rvn-nant of the coeëdenà of Bï aad Sz veshes, namely

g tstme - e-zJzcl = 0, (9.3.5z)

whose most gemea'al solutton is

'mzCYg = j2c (9.3-53)

wkere m is an integer. Substituting in eqn (9-3.50), the upper bonnd forthe nucleation feld is

2.% C =2:r2Hzx u:u 4gr.K - - - + kl + $7,2

. (9.3.54)Mu Ma zlc72

The least negative of these eigenvalues is the oae for wlkic.h m = k = n = 0,Dd for this mode eqn (9.3.54) îs the ume ms mn (9.3.44) of the mherentrotation.

It haa thus been proved that an upper bound for all other modes is largerth= or eqnal to the tnte nudeation Geld of the coherent rotation mode.Therdore, for this case of an inGnite plate, tlze coherent rotation is the modewMch has the Ieast negatîve aucleation feld. It izas not btvm proved thatthere is no other mode which hms t'he sqme nucleation feld as that of thecoherent rotationj and there is no justlcation to the clnx-m in (361) that tàisGculation provu that only coherent rotation e-q.n takeplace in s-ac,h a plate.H fact, it hAA alrevy ben demonstrated in the foregoing that the curlhgmode does have the s=e nucleation feld aa the coherent rotation, aad tldsdegeaeracy leaves some ambiguity a.s to whicah mode q'ill take phce in areal physical situation. The main idea of lixertHscTng the equadons wms to

20O THE NUCLEATION PROBLBM

know 1om wbie.b mode to staz't a numezical solution of Brownss aoa-linereqqaiions out of the very many possibilitîes. For tids purpœe, degenerac'yof the Imcleaïon mode is undeeable, because it allows more thnn onepoôsibility t,o proceed lom. Howevez, in thks respect tke case of an infnitepla* Ls not representative, because there Ls less ambiguity ia any fniteempsoid. Tke aumption of an infnity is problematic anyway, because oftke necessity to assume a periodidty, aleady meationed in the foregoug.

9.4 The Third ModeUsiug m'rnl-lar methods to 'those outliaed in the predous section, h was6mt proved for a spherw and latar for aay oblate spheroid (359), that thecoherent rotation and the curling are tke only pfwmible nucleation modes.Curling 'Ae.S plce above a certe dze, aad coheremt rotation below it,wkere the blrnover from one to the other is given by eqn (9.2.33). TkereOnnot po%ibly be any competition 1om a third modeo because for all othermodes the nucleation Eeld ks more negative thau for thcxse two. The onlyabiguity is encountered in the b-mlt of a.n A-n4m-te plate, for whicll thecurDng and the cohereat rotation tend to the same eigenvalueo as (ILSCIUR'U:!;Iin the previous section. But even fœ that limit, there is no tkird mode) iftke in6nlt.y is approachu from an oblate spheroid with .R -+ x .

For a prolate spheroid, tkere may be a third mode, which will be aamedbncklçng, for lack of a more approphate name- The name buckling was 6m-:applied to a particular model, suggested (&%) together witk'the model for

lin for the'particuzu case of azt infnite crlinder. Its nucleation f eld wascur glater foud (360) 1 be a good approvlmadon to that of a third nucleationmode that was skown to ezst in an infnite cyliader, aad wbiczh was giventhe same namc. Actuallyl this mode turaed out to be always easier thanthe coherent rotation in au infaite cyHder (360), so that iu suck a cylizt-der iere rAn only i)e buckling Mow a certain swize, and curling above it,witkout aay tkird pfwmibility. Tke Grst study (359) of the whole eigeneuespetrnm of a Bnite pfolate spherofd showed that coherent rotation couldbe thc eaaiest mode in some region of size and elongation, but did not ruleout the possibility that tEe buckllng would take over in anotker range. Itdid rule out, however, the possibGty of a lonrth mùde, so that it coald bedvni*ly stated tkat none but tkœe tEree modes should be considered forany prola* sykeroid witk no surfKe anisotropy.

Other limlts on the possible moda were fouad laterj but tkey werestzl ambiguous, until a recent evaluation (363) gave the results reproducedin Pig. 9-2 kere. It plots regions in wlzic,h modes may be allowed in aprola* spheoid with an aspect ratio m aad a radius R in the directbnperpGdicular to tke applied Geld, plotted in terms of the reduced radiusS defned ia eqn (9.2.20). For m K 500 there can only be dther curling orcoherent rotatlon, as shown in the fgurej. and as is the 6a,% for all obhtespheroid? For large,r m, the third mode is not completely mZIM out. It may

THE THYD MODE 201

1-5

CURUNG

COHERENT ROTATION

100/gaFIG. 9.2, The possible nucleation modes in a prolate spheroid with a.u

aspect ratio (major to minor Jt='Kl m, and a reduced semi-minor a'ds, SAdefned in eqn (9.2.20). Only curling or coherent rotation are physicallypossible in the regions so marked. lf a third mode (buckling) eists atall, it ca.n only be in the little quasi-triangle, around the question mark,computed for 'a, = 13. See te-xet for the defldtion of n. Copied fzom (3634.

take place, for alimited size range, in the small trhagular region marked inF'ig. 9.2. This re#on is the best that ca.n be obtained for n = 13, where 'p,)the order of the Legendre polynomial used for the calculation, is essentiallyan arbétrcn parameter. Its choice is only Dmited by the dilculty, whichincreases with 'p,, of achieving a suldent accuracy in the computations.

An elongation of more than 500:1 cmnn ot be reached in practice, and itsstudy is purely academic. Moreover, the bu ' is not very diferent 1omthe coherent rotation, for the inGnite cylinder, m -+ x, with small radi.ibefore the curling takes over. There is some uncertainty in this conclùsion,because of the infnity, whiG is never a 'good assumption in magnetism- ltimplied (360) that the eigenmode of an înfnite cylinder must be pûriodic inthe z-direction. Therefore, the possibility of a diferent, localized mode is

202

still left open, as it was in the study of the infinp'te plate in section 9.3. Butnnll'kx the l-n6nite platej the 1-n4n$ te cylHder rnnnot be approached 6om aknoum solution for a fnite cpsoid. Obdously, if a loeltlized mode'mxleqythe thizd mode, named lbuckling' in Fig. 9-% mus't be simllnr to Gat one,and not to the knowa (36Q) buclrllng in a.a invite cylinder. If no loe-qXeAl-mode efsts in azt infnite cylinder, the bucbling mode ks m-t probablyjust a manifestation of the l'nllntety, aad dx not exis-t in aay fnite ellipsoid.Some attempts to fnd a modelfor such a mode in = infnitecylinderfailed,wlzic,h in prtdple does not prove aaytkin: one way or the other. The mostr-nt attempt (3Gq is wrong because it uses wrong approzmation? Theenerr ttvrn it neglects is (365) much larger thau the terms which are taken

'

into atrount- However, a loMll-g,p.d mode of a similar nature may stzl beptebleo but only within the quui-triangular bounds shown in Fig. 9.2.

At any rateo the mckqt important point is that there zs no other modefor a.n ellipsoid of revolution, and that this statement has been rigozouslyproved and the proof cltnr ot be c'hallenged. There is no pott in tzyiag topostulate any otker reversal mode, Gcause it must lead to a higher enerrbarrier, namely a more negative nucleation feld, tllaa at lemst one of thesemodew unless there is some mistake in the calculauon of the othe,r mode-Nevertheless, there were very maqy suG attempts to look for other modœ,eddently because the presentation of tbis stat-ent in the oririnnl paperswas not dear enough to be understood. The confusion seems to have alreadystarted wlth Fig. 3 of one of the frst redews (366J on elongated particles,whicN put together a sGemxtic representation of lour reversal'modes. Thee of that review (366) did explzu'n the dîFerence, but when this fgurewms copied to many reviews and books, it was used out of cont>t. lt thenled some pxple to believe that there are actually four models for reversal,which may be used on an equal bmsis, not payiag attention to their diflentgeometries. These models are the coherent rotation, bu -

, and curlingymentioned in the foregoing, plus a fourth one rnlled Ianning.

Historicalzy, the magnetization fnrning model was the flrs't aevpt tocalfmlxte any form of a non-coherent maaetization reverml, in order toexplm'n why the Stoner-Wohlfnr'th model did not agreewith expeziment on(m-rtain materials.lt c=e at the <me when General Electric was developingthe production of elongated fne pardcles for what was later sold under thecommercM name of Lodex ma>etw aad it was noted that these particluwere shaped (366) more or less 111* peanuts. Therefore, this shape wasapprozmated (36% by a lin-xr trthna-n of spheresl which touch each other ata point so tbnxt Gere is no exchnange interaction betweea spheres, but thereks a magnetostatic interaction between them. For tMs exq-, a model waaproposed (367j in which the magnetization rotaœ coherently kz tw-11 of the-spheres, but the angle of rotation may be d-eereat for the diArent sphere.This model was called 'non-symmetric fanning'. However, computez's werenot avHable in those days, and computation of the dsF-nt angles by hand

THE NUCLEATION PROBLEM

THE TOD MODE 203

was rather elaborate. Therefore, it was found adequate (367) to study onlyan cppraimation called tsymmetdc fanning', în which there is only oneacgle, with one half of the spheres rotating at that angle, and the otherhalf at minus the same angle. Even this approfmation was shown (3671 tobe easier to reverse than by coherent rotation, which is not surprishgforthis particular geometry. Obviously, the non-symmetric fanning, which isequieent to the buckling mode in a cylinder, must be even easier than thesymmetric fanning, because the enerr is mt'nsmlzed over more parameters,with the symmetdc fanning being a particular case of the iore generalminimization. This mode was studied in more detail for a chain of only t'tvosphere with a unl'ar'a1 Gsotropy whoap easy axis is parallel to the chaina'ds (3681 (which is actually just an additine to the nucleation Eeld ia aJJmodœlz and the nucleation feld for magnetization curling in a chaîn of anylength has been evaluated (344) by a perturbation scheme. The problem ofthe whole set of modes in such a chain of spheres has never been fullysolved, but it seems that the result shottld be very similar to that of anellipsoid) namely that the reversal is by curling above a certain radius,and by non-symmetric Gnning for a smaller radius, witk vezy little ckanceof any other mode. H any case, the reversal modes for an elpsoid, andthose for a chain of sphereq are for dxerent body shapes. They cannot becompared with each other, or mhed together in any other k'ay.

Nevertheless, it was quite popular for some time to compére (36% thevalues tprédicted by each of the known mechanisms of fanning, buckling,and curling' with some experimental results, in order to ând out whichof these Emechanisms' takes place in a give,n experiment. No attentionw.xs paid to the diFerent geometries involved, and actually there was noteven an attempt to defne any particular geometry, even when specifc pic-tures of the sample were aelable. These transmission electron microscope(TEMlphotos showed (369) particles of a rather irregularshape, whkh couldcertaânly not be described as ellipsoids. They look more like distorted el-lipsoids, but they can muc,h Je-ss be approfmated by the picture of a chainof spheres. However, the real shape was not even mentioned in the quasî-theoretical hterpretations, whic,h considered it ae a part of the adjustableparameters, stating, for example, that (370) certain experimental resultsllay in the range consistent with the chaizsof-spheres and prolate ellipsoidmodels' ! The discussed modes were not very well defned either, with moreattention paid to the name than to any sort of a mathematical defnitiopand nucleation Gelds used in these comparisons were, more often than not,those of curling in an insnjte cplï/zder (36% 371q- With this approach thereis little wondcr that too many workers felt free to invent and propose all

. sorts of new reverx models (372) èin addition to the esting models of co-herent rotation, fanning, buckling, and curling', and demanded that thesenew models be treated on the same footing as the tefsting' ones.

A dxerent particle shape can be easily ima#ned to support modes

204

which are dsFerent fzom those mentioned here. But no new shaN wasMsnmed ia thue sindies, and for the same geometry other modes caanottake plaze if properly calculated. lt should lye especially noted that it isnot secient for a reversal mode to bave a lower enerv tlzin the satura/dstate. States with lower enerr do Gst as Kon as ihe applied âeld reversesits direciion, but they are not necœsnn-ly Rcesible, as explained in xction9.1- One shonld be careful to go via enerr minima on a well-deîned path,and not to use approm'mations, which can lead to large errors, as they havein these modes. Thee new modes esseutially nAsmmed the ume chain ofspkeres, but considered a vagaenes about the geometry as an excuse forpoor approvlmations in the calculations. The kuass-curling' aad kuasi-buckh'ng' (370) were particularly ill-defned, and were hardly more Ganjust names, it being svted that tcalculations appropriate to thee caseare ve.ry dilcult'. The Knove.l reversal mechanism' (372), named 'êipphg',could be the same as what used to be rztlled non-symmetric fanaing ina chaîn of spheres, if done properly aad with no extra approimations orinconskstendes, suc,h as a total thickncs, T, of tEe TGzlly developed wall' inthe chaiw which is allowed to be largear than the total number of spheres.

ln an invited talk at a conference (373) I tzied to point out these dLs-torted concepts, but Kncwle at lemst was not convinced by ihese arrz-ments. Ee publsshed (3741 a lreply' whic: stakd that the i.r shapeof the partides invalidated all the reults for an Gipsoid, and allowed %,'mto legislate curling out of efstence, aud to choose other mode at wi11. Therecder will hopefully understltnd that this approach leads nowhere. Thereare more appropriate studie of cltn.ims of spheres wiih anisotropy (375j) orof spheres wllich are cut before joining togeoer (376) <) that they touchGCX other over a rather wide area,, and not only at one point, and thusresemble better the pe=ut-shaped pvticles. There ls also a theory for anhlu'n of disks (3711, and one for a chain of oblate spheroids (378!, whic,h areeven better approfmations for tbsK skape. Yd none of tke.m has encoun-tered any new mode, aad they a11 come back to the 'old' moda. In orderto condude tbis discuHon it will only be mentioned that even the angalatdependence which was so emphaaized in (374), as we,ll ms that of (3711, waslater azcouated for (379) by a completely dl-lerent approach. The data ofKnowle were ftted to a Stoner-Wohlfne: model, with a cubic maae-toctystxll-me anisotropy superimposed on the shape anisotropy of a prolatespheroid, and whe.n the demaretizing feld wms introduced, it led (379) toazt texcelleni agreement'. Other possibilitie (368, 380, 381) have n.lgn beenœnsidezed. R doe happen quite often (3821 that the same experimeavldata ft dferent theories.

Tc NUCLEATION PROBLEM

9.5 Broml's ParadoxThe mnsn rexson for the tfatilel search for other reversal mode is Gat inspite of all the rigour in the qvaluation oî the nudeation felds, the results

BROWN'S PARADOX 205

do not agree wîth eoeriment, in particulaz for the case of bulk matedals.Mthough the nacleafon feld is a tàeoretical concept which is aot usuallymemsured, it ks not evem necœsary to continue the calculation beyond themuclGtion point in order to see that this theory cauot apee wit; etper-iment, and that something is basicatly vrong with it. The nucleation îeldE= is deîned 5n section 9.1. :: the feld at whic.k some ckaage fust s'tcr.tsin the previously saturated state. The coerdvity (or coerdve force) H< isdefned in section 1,I as the feld at which %x1f the maaetization has beenreversed, whkh at any rate means that a tvge change haq already ocrmvrMin the previously nturated state. Therefore, Hz can only be renrhed a?.ergoing through Jlk in the sequence described in seclon 9.1, or tkcsy eztn beat most tke sxme feld in the e-qme when there is a big jump of the maae-dzation at the feld H=. Noting that the deânjtion of Hn is for an appliedfeld that stuts pœitive, :zd goes tbrough zero to negative valuu, whileHc is deâned ms a positine quaatity, the above statement means Hc k -H=.

Suppose ârs't tllat the fezromaaetic body is a large ellipsoid. Smallparticles will be discussed later in this section. The main conclusion fzomall the studies in sectîons 9.2 and 9.4 is ibxt for a hrge enough body, thecurling mode takes over, whether it îs coherent rotation o: budding for thesmaller radii. Therefore, according to eqn (9.2-31)

2A CeS k ----1 + - NzMs.c x azvs (9.5.55)

The right-haad side mqy be negative, when nucleation occurs already atpositive âelds. For example, for '-wm at rxm temperature 2Kï(Ms =

560 Oe, aad 4xMs = 21600G. If the elpsoid is a sphere, Mth. Nz = 47/3,the frst aad the last term of mn (9.5.55) add up to -6640Oe. The middle1term is positive, but it is ceriminly negligible for a sulciently large R, andthe z'ight-hand side of mn (9.5.55) is negative. In such a case all that theixequality states is that the posiiine Sk is larger than a negadve number)whic.k is an empty statement. EFectively it means that the calculation ofH= is inadequate to tell anything aboat S< in such a cmse. St - from alarge positive feld, a reversal already nucleate at a podtive applied âeld,bdore a zero feld is reached. It is then necessary to solve the non-linearBrou's equations for smaller positive felds, after nudeation, and followthe solution down to negative felds ti11 -Sc Ls reached. No statement aboutthe theoretical value of Hc is wlid lxfore thls calculation is carried out.

However, if the iron cystal is an elongated prolate spheroid instead ofa sphere) Nz caa be much smazer (it tends to zero for an infnite cylinder),and the whole NXM. tprrn noy beome negligibly small. ;.a this case theright-haad <de of eqn (9.5.55) is positive. Since the middle term is positiveanyway, it eztn be stated that for ve,ry elongated iron bodies Sc 2 560 Oe.For iron whiskeest which are very elongated paztida indeed, with dixmeters

206

of severi lm aad a length of the orde.r of 1 cm, the exwrimental value fozHc is nsually 0.1 Oe or less, wllic.h Ls a ve.ry large discrepaacy. lt ks lœownia tàe litaratuze as Btomn's Nrtotrl or Brown's c-rcivity paradox.

lroa is givea here ms a zepresentadve ok a chss of mat-n'xh wkic.lt .6called :o# magnetic mzedcl: and deoed as materials fœ whic.h .R% <JrM; . The othez exi-me is matezials for wlzich Kz >. 'ZMI, and thks clmssis Hown by the name of Mrd mcgnedc materials. There is ao 1aw of naturewhich preveats maiarials fom being in betweea these extreme cases, butsuck matezials are not behg produced or invœtigated, beceause they do nothave aay practicat appicatioa. Soft ma/rials are used where the coerci-vit.g is preferred to be as small as poggible, e.g. in motors or trusformers,where a hlgk permeability and low losses aTe zmuired. Hard materials aremsed in applica/ons whic.lk require the magnetizatioa to be fxed for a loagdme after the czystal ilM been magaetized, such as in permaaent magnets.Recording materixlK, sueah as 'PFeZO3, aa'e i!l fhe latter categoryp but forthem it is prp-f-mad that the magaetizing feld for writing the data skouldnot be htge. 'Thereforq they are made wltlz a Kï wlzich is rathe,r large,but not too large. Such materi/s are sometime rderred to ms semi-hatdacgnedc mcfedcfxs. '

H hard mateziab, Browa's paradox is even more outstaading tbxn insoft oaes, becauseit appzes to any elliped, aad not onlyto elonoted ones.For ev-tmplej ia R-tpezzozâ at room temperature 2A'1/Ms = 16 600 Oe, andyrMs = .1500 G. The largœt possible *ue for Nz in any cpsoid is 4r,fœ an inHte plate with the feld perpendicular to the plate. Even forthks value, and just noting that the middle term in eqn (9.5.55) is posiîivewithout chenldng what its mlue may be, this ixeqnxlity >ys that Ho à12 000 Oe. The experlmental smlue for Beelcow partides of the ordez ofl>m is 3 000 Oe. Slmonr discrepandes are eacoune in practically anymagnetic mateM, whea the crystal size is hrge eaoul.

It should be particnlnmly noted that this discrepancy, or paradcx, caanbe fozmulated without any of the Yculatioas in the previous secdons) aadwould have applied even if all that algebza was wrong- The value of themiddle term in eqn (9.5.55) has not even beea used here, except for itsbeing positive, so that all tke details of the curling mode do not exte,r tàergnment. At least the same discrepancywoddhaveapplie if the coherentrotatioa wms the easiat modep because it is in the term with Jt% wkic.lt iscommon to c!J tevetsaz modes. Otker modes may even Tnnlw this dirvfnre-pancy worse by having additional terms, such asthe Nz term iu eqa (9.2.10)for the coherot rotation, but the term with Kz is certainly always there.For thl'n reason, inventiag new modes caaaot help remove the paradox)ev= if atl the discussion in =tion 9.4 was wrong. Actually, tke problemwms already Hown b6jor6 all the foregoing calmzlations of nucleation felds.Mready in 1945 Browa 13831 had noted that the barrier for nucleation ofany sort of a reversal is ai lecast tkat of the anisotropy ezezut wbich is

'PHE NUGUATION PROBLEM

BROWN'S PAEADOX 207

the Erst term in eqn (9.5.55), just because the exeunge aad magnetostauceaerpr terms are alwan positive. Even this nnt-sotropy œm by itself is too

' large, and the e-xpersmental coercivity is considerably smaller than it, inall bnlk ferromagnets- Moreovezz even the valaœ of the coerdvity need notbe nKtvl in ordc to rmllh'z- the paradox. Equaïon (9.5.55), or already eqn(9.2.31), implies a negaiine nacleation feldj but domains can 1)e observedalready in zero applied âeld (see sedion 4.1). It has ben shown in ution6.2 that the estence of these domMnR is favorable energetirAlly) but alowe.r enera is not a sulcient conditlon for the domalns to enter. As ha:alzeMy been uplznm-qed in section 9.1, the ezstence of a lowez-enerastate is not suEcieat Tor the system to be able to repnh that state.

. The remsons for this paradox are rather well understood, qualiiaiinelk,and will be listed separately for hard and for SOA matlm'm. Eowever, Moregotg 1.nt0 these details it must 1)e emplxized that the discepandes aze txlarge to be taken ligbtly. Until the theozy is modise to take into accountqnanutazively tke Gects whic,h cause thts paradox, everything discussedin this chapte.r eAn only be applied to Ene partîcles. None of this study ofnucleation can serve a=y usev purpose when it comes to czystals whichare large enough to suppol a subdivision înto many domains. For themit is possible to get away with a theory which iaores all of this càapte,r

d tlte question of how domains enter into the cystalz' and takes it forgrante that they are there if their estence reduces the enira. Theorieswhic,h only compare energies of vadous confgurations, such a,s the one insection 6.2, or that of domain wall stmtdures i.ll chapter 8, work very we.llin this refon, aad eztn be used to interpret all xrts of ' %ta1 data.A particularly nice t?y'n.rn p1e is the shape of tlze rvfwtxlled N&1 spikes, whichare formed (384) near non-magaetic inclusions- Once their general shape isxuumed, an eaera mlnlmlzaxtion leads (385) to a.tt the 6ne details of thdrstracture, with a perfect R to experiment. Such theoria are still beingapplied (=) , aad they are actually inevitable, as long ms the present theoryc=not be extended to take care of the defects whic,h will be spfvq4ed inwthe followlgu Still, this situation does not jusGy discarding tMs càapteraltogether, much less discarding the wkole theory of micoma&etiœ, ashas beea suRested on dubious goulds, discussed in (382). The nucleationtheory does agree with expeziment for small pardcle, and the modiîcationswhicà are nec- to mnlcm it >ee <th experiment for largez particlesare knowa in pdndple, aad may be worked out in detail somekime.

9.5.1 Hard JrflterilldWken opticaliy transparent plates of BaFelzozs are saturated in a largefeldp aad the feld is then ruuced, domasns appear already at a positiveappEed Eeld. Eowev/m it has been noted (38% that these domains do notappear anywhere in the Mmple. Thez sem to emerge radîally from a well-deine (nucleadol centre' whicà îs always at the same spot in a given

208

crystal, for vadous cyclîng of the ield. ln some crystals, no domains wereobsened at zero applied îeld, aad they only appeared up to seveeral hours(:'&) nfl,er the feld kad btam switched of. In some cases, domains nucloated only after a negatve Geld of -1000 to -2000 Oe was appied :389).Obviously, these observations seem to indicate that the domes nudeatebefore the theoretioal value is reached only at those poin? in the CY:aIwhere there is some sort of a defect, whiclz may be, for evpmple, aa impurityatom or a dislocation. The nat'um of thee ddects hrm not been determined,except for one case in whicx tke nucleatlon centre could be identled (389)with a craclc in the crystal- Similar nucleation centres were produced (390)iu MnBi fllmq l)y prickng them with a non-magnetic needle. Some domainsnudeate (3911 at the ehe.a of a plate,

These experiments may meaa that the peded parss of the crystal obeythe nucleation theors and that nothhg would have nucleated if tt were notfor these imperfect spots. Of couzse, oncc the domains nudeate at one ofthese centres, there is no extra enerr barrierj and it is ver.y eas)r for themto spreH all ove,r the crystal, when the state of subdivision into domainshas a lower enera thztn the saturated state even for the perfect parts ofthe crystal. A model was proposed (392) in whic.h the nucleation centrewere assnme.d to be dislocation lineq and their efet was assumed to be ahigh local stres that could eedively be taken as a local Jouednp of theanîsotropy constant, K:- Nothing specifc can be done witllout atleast someindicatioa of $he amouat of this reduction, aad tàe size over whc.h it mayGxtemd (3j32, but these parameters are not lœown. It is x1m not possibleto know what tle domains actually look like ia the early stages of theirformation, and an attempt to study tikis stMe (394) in one materln.l couldonly report that the initiat growth of the domaîns pro-ds tx rapidly forobservatiomFor these remsons, suc,lz a calculation, or its modifcations (395)396J, can only be descxibed as a semimuaatitative evaluation- lnstead ofthe dillocation lines, the defects may adually be planar (396, but in dthercase it sems (392) tkat su& a mraztunism r'lm resolve Brown's paradA inhard materials, at least for partkle which are aot much larger than thesize at which domains start to be energetically favourable, atthough a morequantktative theorr is still needed. There are also some e-X.IIe-IiII'eIA on

- - and ann-wkh-ng hard matpvlnln, whose resul? are g112, 113, 114, 398)in qnxlitative aveement with tltis picture for the role of dislocations.

Foz much larger cystals, the role of dislocations is reversedj and thereare both theoretical (399, 400, 4011 and experimental (402, 40% 404) in-dicadons that the c-rdvit.y of bulk materinls hcreases with increasingnumber of defects) probably because they hold the domaia walls and donot let them move fzeely when the fleld is nbltnged. There have bcea sev-eral attempts 2405) to separate the mechanisms of nucleating a domah aoof moving its walà but the diference has not bee,n (4œ) very well œtab-lished. Or, as coacluded in a review (4Wj, Smore work is needed to make

THE NUCLEATION PROBLEM

BROVWYPARADOX 209

interpretations unambiguous'. The problem is particularly compllcated byexperiments whic,h do not start with a suëdently large applied feld fordriving the domains away, and report measurements whic,il are actually mi-nor loops. Sometimes they are presented as suc,h (403), but sometimes theyare not, as in several examples listed in (373, 382, 392), To repeat just onecase, in certain MnBi crystals the domalns completely disappeared (408) atan applied feld of 5000 Oe, and reappeared when the feld was reduced toa smaller, but still positive, value. When such a crystal was once put in afeld of 20 000 Oe, the domains disappeared and never appeared again withany cycling of the feld. In some other cases, it has been stated that thetnucleation feld depends on the value of previously applied positive feldand t;e crystal imperfecuon' (4091, so that the nucleation thus measuredhas obviously nothing to do with the nucleation as def .ned in section 9.1:or with Brown's paradox. Similar obseneations (410, 411, 412), and othersdted in (407), show that many experiments do not start fl'om saturation.On the wholej the magnetization reversal in bulk hard magnets cnn only besaid to depend on crystalline defects, which are not included in the theory,and that Brown's paradox wûl be resolved when they c,r6 lduded.

9.5.2 Sojt Mstezicglf a long iron whisker is held in a suëciently large magnetic f eld, and anopposite feld is apphed to a small part of it, it is possible to study thereversal of tbm part of the whisker, while the rest of it is held saturatedparallel to its long aMs. By picldng the signal 9om the reversing part, itis possible to determine the feld at which the reversal just starts, namelythe nucleation f.eld, f or dxerent pacts along the whisker. This experiment(413) and its later modifcation (414) obtained nucleation ields which werequite close to the theoretical value of -560 Oe (for a very long iron coestalat room temperature) at some partks of selected whiskers. ln other partsof the same whisker: less negative values of Hn. were measured, obviouslybecause the crystal was less perfect in those regions. Unlike hard materials,for which the nature of the defects at the nucleation centres is not Mown,for soft materials it haz been established that reversed domains nucleatewhere the surfxce is rough. The f rst demonstration (414) of this conclusionwas an electropolishing of the whisker, which resulted ln a complete changeof the whole nucleatioa pattern. A more direct proof was aa observation ofthe surfxce of the whisker by an optical microscope- It (415) showed a goodcorrelation between the volume of the surface defects and the dxerencebetween the theoretical and experimental Hn in that vidnity.

Surface roughness must be important in any ferromagnet, but i.n a softmaterial the magnetostatic enerr is particularly large by defnition. Thereis thus a Iarge efect due to the surface charge created at the polts wherethe magnetization of the saturated state is not parallel to the surface, whichmust be the case where the surface is not smooth. This charge gives rise

2lû

to a demagnetieg Eeld in that re#on, whie,h reduces locally the energyharrier, thus atlowing domaâns to nucle.a* there. And it is quite easy to beconviuced (372) that a local 1A111 or a local valley hms approfmately ihe sameGect- A Kim5lar demagnetization, due to a similar sMrfn ce charge, shoutdalso occm near voids and knclusionsinsidethe crystal,whicà llave also beeademonstratH (416, 41% to intmd with walls. Nucleation at such internaldefe? is xlg.n possiblw espem'zzly in less perfu samples. Even in whiskers)which are particnlxrly good crystals, there were xme cmses where a localreduction in Wkl could not be mssigaed to any surface imperfection (415)and must have been due to an iuternal void. The opposite never occ=ed,aad wkenever a major snrûce defed could be seea on at lemst one of thefour snHves of a whisker, therewas always a rnînirnnrn ia Iskl there.

This total dependenœ on the fne details of the surface in a large crystalmust seem straage at frst sigkt, especially to somebody who is used tot%înkiag in terms of the theries in the frst few chapters of this book. tnthose theozies, a suedently large crystal (and sometimes even quite smallones) are just assumed to extend to ''nGnlts with no smface at a11. Thelnatzzrap approltn% is that the surhe.e can àave a strong eKH onlyfor smallcystalw but it is Sexpected to be.l= impoztaat, the lazger the czystap asIwrote (4181 in one of my earlier papers, before I understood the nature ofthe problem. The point whicl must be remembered is that for spmdentlylarge custals the muld-domain sute hms a lowe,r enerr tha.n that of thesinglodome one, but theoretically the domains cannot nucleate before acertain negative feld is reached, because of an ener bxvra-c on the way.Once t%5K bxrrler is liftad at any point in the crystal or on its surhce, itLs easy for them to propagate all over the crystal, as seen exzmm'mentally(419), and Gcause thdr estenee reduces the enera eve,a itl the pezfectpvts. Therefoa aay mewwx-ment of the whole m'ys'tat will measure thenucleatkn pzoperty of the morst poht (3821, ï.e. the point at which thecrystal is Gethest fzom behg perfect. The Xect is the same as in pullîng aehnln, with a stenzh'ly increasiag forœ. The whole chai.a breaks at ihe forcewlzich Ls suEdeat to bm.;tlr its m-kest linlc, even if a,ll the other Bnks aremuch stronger. Tke only way to mecure the properties of other linlcn isto pull them one at a tîmey while preventing the others lom being pulled.And thb only wa,y to End the true nucleation îeld of the perfedly smoot:wMsker is to measure one region at a time, while preventing the domnsnsfrom entering the other parts, by kœping tkem irt a saturating feld, as isdoae iadeed 1413, 414, 41SJ in the uperiment of De Blois.

1.n this sense it caa be said that the ex-periment of De Blois proves thatthe nucleation theory doc agree with exwrHent for the perfect crystalsassumed in the theory, andthereis no paradox.For l%s perfect crystalsl thetkeor.y should be modïed to take into Mcount the eeect of imperfecwdonGwlzich has not been done yet. More details were revealed in two extensionsof this etmeriment. h one (420) a local feld was applied to dferent regions

TM NUCLBATION PROBLBM

21l

of a thin flm, and the sham of the develophg dome was obsewed. Inanothe,r (421) whiskers unde,r stress wm'e studied by the same tchnique ofDe Blois, and the nucleation feld at tgood' points was Bund to becomemore negative with inœeaing str-, whic,h is equivalent to iztfrrpnm'ng theanisotropy constant, .&i. There is a frst theoretical step towards a betteranalyis ofthe data in the 1= perfec't pats of the w%lqklr in the ex-pen'mentof De Blois (422), and a stadsdcal corrqlation (4234 betweea the pzobabltyof fndsng a defed and tNe values of Hn measured by De Vlois. But noneof these studies haz been cxrrled Z'kr enough for a quaatitative theory ofimperfed crystazs.

As is the caze i.!l hard materials discussed in the foregohg, there arealK theories for soft materials whicà iaore nudeafon, aad try to analyse

' ental (Ia.'t.a oa H'zFere'at memsurements, in particular coerdvities, ofthe crystal as a whole- They usume that the domains are azre-ady there,and consider pinn'lng of theiz walls by crystcne defects-For these thKriesthere is not really much Hilerence between llard and soft materials, exceptfor the numerical parameters wllicà aze used. Some tîeories still use theassumption of one wall pinned to one defet (424), while others conside,rthe statistical aspects (425, 426) ok many defects, randomly distributed inpnnh domain. Vazious models have b- proposed (427, 428, 429, 430) forthe phniug. They all Mse rather rough approvimadons. '

nœe is one point on wlzic.k the experknent of De Blois' cannot #vea clpm.r namRwer, and whicà therefore remains obscure. It Ls the question ofwhether the edge (or tip) of the whisker, wllich cannot be accessed by theteGnique of De Blois, behaves d-lFerently than other regions. This questionhas alreadybeen iscussed in (392) in conaection with some suggestions forresolving Brown's paradox by the argument that real custals are neversaturated to start wîth; and some uns@m domaîns reme npltr the edgœ,w' here the demaoething âeld (for non-ellipsoidal shapes) is veU 1ar&. Of(xptme, there is a large nmount of evidence, some of wlzicà ha alreadybeen meafoned in the foregohg: that the feld used in many experimentsis not suRcient to saturate the sample, and these published results confusethe l'aqne. But there are those who claim that no sataration is possible inJrincfpl: for some bodies (such &s a prism or a plate), with a sharp coraerat the edge, whieà is a diferent matter altogethemMy argument at the time(392) was that on a,n atomic scale, a sharp corac doc not have any moremeaning than a zounded one (see Fig. 3.1). Theareforq the approfmationof an ellipsoid is at least as good as that of a prism, aad a saturation ia a4nite feld shoald be posdble in prirciple for real particlew although highe,zâelds th= are usually conddere admuate may have to be used. My view

. is e111 the samej but the opposite is just as legitimate; aad there are thosewho prder to consider a pHsm, which lAke,s an ''n6mlte âeld to saturateand for which there can be no audeation tsee also section 10.5.3). On 11H*basis, there ks a model g431) for domnins that enter fzom the corne,r of a

BROWN'S PARADOX

THE NUCLEATION PROBLEM

plate, even in Mrd materials.

9.5.3 Small Parlîcte.d .

Al1 the foregoing Gmlaaatiozus of 'mltk the theory does not ag' ree with mostexwrimeents do not changethefact that thetheoTyproented in tlzis chap*ris not useful for mœt practical cases. nerefore, this t'heory wxs amdderedfor a long time to belst a rmriodty, which could at most interest some puretheorists, or may bc appliM to extzemely unusual ldnds of experiments.And in spite of all the great hopes of Bzown and others in the beginning ofhaving one theory that can exple everytMng, it is undemiably a completefaa-lure for bulk materials-lt should be obvious 1om the above analysis thatuntil a big improvement is incorporated, this thmr.g can at mœt be appliedto small particles, below the value for which subdidsions into domesreduce the total energy. If the nncleation Ls aot that of reversed domains,the mct shape of the czystal becomes less important, and b0th the KeL'tof costalllne defec'ts and the probablty of the Gstence are vezy mu&zedaced. For sach cases, the nucleation theory has a chance to work we11.

H a way it exn be said that it works indeed in a relaévely narrowsize region of tsmall particles'. This reon Ls bœt dened 1om a plot,sucà as Fig. 1 of (190), of the remanence and coercivity ns. pn:rklcle size.Thesg properdes lhn.ve 1ow values due to superparamagnetism in the smallerpvticles, aad due to subdivision into domes in tEe larger ones, with amemum in between. H the vicinity of this maximmm, nudeation theoryupapxy wozks, ind since the smazest particles are elirnA-nnuted, it meaas inm*t cxq- that tie theory of the curling mode ayees with experimentaldata. Examples of such agrœment have been listed in redews (270, 35%37% 392), and it will only be repeated llere that the coerdvity of verzelongated niekll particles was found (432J to be qaite well approfmated bya D-.ftr ynction of VR2 at two temperatures. A linear fundion of 1/12wms also observed (4335 in tEe coercivity of alumite, although the pnrkseleswere not ellipsoidsy and even in cnbes of cobalt-doped 'pFeaoa :4341. Thelatter Et was ori '

y praented (434) as dperhaps fortuitoash, but it nowseems to be a real part of a pattern. Tkis example, acd others, prove thatthe iheory works on the wkole ix tMs size raage, au.d may at most needsomeslight modifcations, when theMer details aretakeainto account. Thedata suggoty as a zough citarion, tAat the theory works 'well if the middletarm of eqn (9.5.55) is larger than the frst one, aud breaks dopn wheu thefrst term becomœ large. This rule ks demonkzated by the experlrnentalcoercivity (434 of whislrom. It is clœe to the theoretical =lue for curlingnucleation at small radii, bat when the Xz term becomes dominant, the co-erdvity keeps decreasing, orders of magnitude below the theoretical value.

However, such a comparison of Hs with Hc does not really show anagreaent, and it is morc aczturate to say that in this size re/on there is nobig discrepancy tha.n to say that it fts. The Mreement (4:$6) betweea other

BROWN'S PARADOX 213

propertia, calmzlated 1om the curllng mode, aad the experimentaz data,is n1m sai-qualitative. Sucb fts used to be considered good enough whenit waz discult to make small particlœ, and when expeziments were done

.on a large number of particles togethez, wllic,h involved the crudal Otorof size distribution. And even with the moze recent contzollM dispersion(43-1, thue ks stzl the dimculty of interactions among the particlœ. Thisproblem hms never been solved, and is still studied (438j for the case ofa regular array of particle, or by cerinn'n averaging (4395 schemes. Theformer theory should certainly be compare with expezimen'ts g440, 441)on such an artifdal, regular array- But it is unrealistic when interpretingexperiments on disordered powders, ms is deemonstrated by more rigorouscalculations (44% 443, 444) on imo iateracting dipoles) wbich do not ftthe local-feld concept- Suc.h calculations can probably be a goV baés forstudying particles wkc.h have (445) a;n odd shape, but they caa obviouslynot be extended to interactions within a random ensemble of particle.

Nowadays, this problem of interactioas enn be evaded instead of being' solved, because more and more mfuxurements are performed on a singleparticle 1369, 446, 447, 448, 44% 450, 451, 45% 453, 454, 455, 456J. Forthese erperiments, fàe sqrni-quantitative compadson of H.n with Se is notgood enough any more, an.d it should be possible to tzy a more detaâledcomparison. In particular, a good thKry is aeeded for the coerdvity inKipsoids, preferably somewhat distorted elliyeids. It should l)e note thatthe ccerdvity is not only an import=t pnmmetez h i? own right. It ks oheoî t;e 'lmry Jet12 parztmeters from wkic,h tke wkole hysseresis curve can l)e(457, 458) constracv.

The ori/nal id% was to conthue with the solution of the non-llnearmuations, on- the nudeation is determined, and ît is possible to identifytlle brazmh on which to proceed. It q'as not done because the di>gr-mentwità experiment for bulk materials gave fhe impression that there was nopoint in continuing: when already H,% wms wrong. Small pardcles were notavailable exmerimentally at that tkme, and the witole f eld was consideredimpraztical- The Omebnnlr was whe.n the size of recording particlœ becnmesmall enough, and the o1d theory suddenly f tted many experimental data.But at that time many of the original papers w&e already forgotten, ormisunderstood: and the new theoretical approwi did not proced the wayit could have. There'wu too muc,h èfort put into snding other modes thatwould ft better: and too little efbzt put into necessazy modif cations in thecttrling mode that œuld be appliH to real particles. And in particulv thereis still no attempt to do what Brown meant io be the next step to start with,namely to fnd out what happens c/fer nucleahom Jm. the begkming, thezeis no change while the feld Ls reduced 1om a large positive vallze, throughzero, and down to a certaân negative value, when something nucleate. Aherlâat, the linear equations are not valid any more, and the next step shouldbe to solve the non-linear equations for more pzgctirc felds, and follow the

214

rest of the magnetization hysteresis. In tMs solution, the one that is to bechosen oui of m=y possibilities is the one whic,lk teads to the nucleationeigenmode whea H= -+ Hg%. Rxperlments are ready now for a good theory,

s'

.. '

but this part is s'till .

An attempt to do just tut part (459) for a particttlar case missed thernnsn point, and after fnding the nudeation feid, H=, these authors solvHthe non-linear equations for IS'I < ISkI. They 24591 found that tàe eneoof the curling mode i.!l Qzis r=ge of feld is larger thaa that of the nnlf' ormlymMnetiznl statev wilic,h Ls not surprisinp M it means is tbat nothlng willhappen in the range lS'I < ISkl, wkich is essentey thc desnition of thenudeaéon f eld in section 9.1. They should have solved for H < -tSk!, inorder to 6nd out what happeas c/er nudeation. There are iadeed in theliterature some works in whii the eneo is calculated for IS'I < ISkI, butthey (460, 461) look for the energy barrier, whick is a deeren.t problem,or try to Sad a local mode, as mentionH in sedion 9.4, which is also aaieewmt problem.

A xludon of the non-liapm.r equations wms only tried for an inf nitecylinder (360), whicb is an atyplcal cmse. H that caeit was found that therewas one jump fzom mturation along +z to one along -z, and Hc = -Hn.For a sphere (462) and 4or a fnite cylinder (463, 464) thea'e are only someapprczimations to the krue cuzling mode after nudeahon. The behaviourof aa ''n4nste plxo wms only studied (465) at the frst stage after nucleation.Nothing bpn been done for a more general ellipsoid, and lt is still needed.

'1'HE NUCLEATION PROBLEM

10

ANALYTIC MICROMAGNETICS

ln this ckapter aad the ne.u one,mious topics in micromagnetîcs, outsîdethe nucleation problem, will be decribM. The subddsion iato Annlyticaad numeriY stndies in the two Aapters is rather artlcial and quite ar-bikary. ln most cases thae is no real distinction between analyticsolutionsaad Immerital ones, and m=y physical problems use a ml-vkzzre of b0th..Nevertheless, it seems desirable 1om a (Iidvdc mint of view to keep themas separatû cbaptars.

10.1 Ferromagnetic ResonanceThe bmsic equation g

- this rvnance is eqn (8.5.48), or rather oneof its modifcadons as either eq.n (8.5.50) oz eqn (8.5.52), because there isiways damping in rpml s. The Gperimental setup, always hwolvesaa applicauon of a larr DC feld Jzk whtch holds the magaeth. atloa almostparallel to its direction) z. It mp--avm that the components perpendicular toz are rather small) and may be taken to a frst order only, ms Ls the cmsewiththe nudeadon. Besides the DC feld, there is aISO an AC feld at a givenfzequency) uz, whîc,h dticldes' the magnetization into a peziodic motion atthis fzequenc'y with a small amplitude. One is tàen looldng for a reonanceof ih% motion at the lequency u7, when the applied DC feld Sa passestàrough the appropriate value which corresponds to u? being the frequencyof one of the natural oscadons of the system.

If the AC feld is sinusoidâ), its time-dependence eztn be expresed bya factor cï*'z and tke same factor e= then be inserted into the steady1state solution of mz and my. The linpxn'zAtion of the equations for mA11znz and my Ls mathematicaEy the nme ms for the nudeation problem iasedion 9.1, for the sâme vqnvn pdons used there, namely that the sampleis aa ellimoid, aad that the feld Ls applied along one of its major axes,wllich is also azz emsy n='n for either a uniazal or a cubic anisokopy, etc.For this case, and when the damping of eqns (8.6.60) or (8.5.52) Ls omittedfor simpidty, it ks seen that the equations of motion for the amplitndestmxmely whea the fMtor e*f is omitte, are

C z 2Jf1 i;/ 8UïmV - + N>MS - m rzz - - zn.y = ,Vs Ms Jo 0x (1 0.1.1)

' aad

216 ANALYTIC MTCROMAGNETICS

C ; zffz ïte OUv=V - + N.M. - Sa ms + - mx = , (10.1.2)V; Ms 'M 0y

where all ihe notations are the same as in section 8.5.The bounde conditions arethe nxm e as in the case of nudeation, and

on the whole the nudeation proble,m may be regarded as a particulr caaseof the resonuce problem, for the particular value u? = 0. There is, however,one big dl-/erence in that the nucleaïon has a physical meaaing only forthe mode which hms the largesi eigenvalnw a: discussed in chapter 9. J.uthe case of the resonance, @J! the mMes caa be udt.fvl in pzindqle, ifthe conditions are right, and the efstence of one mode do% not ertmmnteany of t'he others. And mzmy dl#erent modœ have lndeed been studied

- entally in the snme sample.In spite of the shm-lxrityy reonxnce modes have been studied without

payingmqch attention to the relation to the nucleation pzobleam Yfoze andafter this point had been discussed (341) by Browm h some ways, the theoryof ferromagnetic resonance is more general th= the equations given here,because it somefsma iacludes other terms, such as a s'urhce anisotropp e-pin (158, 466, 467J. R n.1M has to take into account a more general form ofMax-well's equations lba.n is used in this book, because the dynxmx'c Xecisof eddy currents and skin depth are important (468) at the higlt frequemciesused in these expersmenis, whereas they aze negligible izt suuc nudeatiomH mosi cases, however, the geometv is Bmited to Gat of an inqnite plate(4691 , for which the solution of the diferential equations is Omposed ofsinusoidal variatiö'xus, ms in =tion 9.3 here. Beides ihese sinusoidal mod%(4701, the theory of resonaax zvwcorizes only the coherent rotatioa modeAown 'ms the tttniform mode' iu tkis contexi), in whîch boih mz aadmv aze constants. The non-coherent modc in a.n ellipsoid, known ms themagnetostaiic modest are only studied without the excbange enera (471),by writing C = 0 in eqns (10-1.1) and (10.1.2), mnn-ng tkem algebraichstead of dferential equadons. This approimaiion is Justifed as long msthe ex-perimeatal Rnmples are rather largel and a rough estMaiion (472)Aowed that the neglected exchange term was indeed nesigible for the sheof spherœ IXS?I tiken. However: smaller sphere were made (473) later, andihey were found to sqppoz't new resonance modœ (4'F3), wkich dewnd ontàe particle sizet and whic.h have not been observM in larger .parkkles-

In such small sphez%j' the exnbxnge energy bccomes dominant, aadit is necessar.y t8 xlve the difeential equations izt the same way aas inchapter 9. As a ftrs't step, this problem wa,s solved for veoe small spheres,for which the magnetostatic term is neglkblecompared with the exchnxngeenergp For the case of a cylindrical symmetrs the whole problem can besolved analytically, and the result is (472J the same as the curling mode. hparticular, in these modes, for which the n=e emcltange rrlod> has beensnggeted (4721, there is a term propordonal to 1/.R2 in tke resonance ield.

FITLST INTEGILAL

lt should be particulatly emphashed again that higher modes ca.n also beexcited, and that their resonance can also be observed, unlike the case of thenucleation feld, for which only the least negative eigenvalue has a physical

xmeaning. For the usual sinusoidal variation in a plate, the higher modesare the harmonics, with integral multiples of the lequency. For the curlingexchange modes, the higher modes are obtained from the larger roots ofeqn (9.2.26), which should make them emsy to distinguish.

Recently, resonances that may be assigned to such roots were observed(474) in small spheres, but the size-dependence waz not that of kjRI. Sur- .face anisotropy is a very lîkely cause for the dference, being demonstrated(475j to be able to make the theory agree with experiment. If it is thereazon, it must also be taken into account in the nucleation theory. Theresonance theory, however, is not suëciently developed yet to rule out othezpossibilities, epecially in the intermediate size range, for which 30th mag-netostatic and exchange ener#es may be important. H this range, someexchange modes may be mix. ed together, and less easy to tell aprt. Also,the modes without a cylindrical symmetrs which have not been calcu-lated, may overlap the other modes in the same experiment, espechlly atstill smaller sizes, when a curling coniguration involves a vezy large ex-change energy. The theory of these modes has also been extended (4761to include, for exnample, damping, which is lefi out in eqns (10.1.1) amd(10.1.2). But this extension did not address tlle mode mifng, or the casewithout a cylindrical symmetry. Before suldently small particles could bemade, this theory was only a mathematical exerdse. But now that suchpazticles are available, these gaps in the theory should be investigated.

10.2 First hdegralIt has been mentioned several times in this book that the mssumption ofa one-dimensional magnetization confguration, in a.n infnite crystal, isquite risky aad may lead to serious errors. However, this assumption ismade anyway in many calculations: whether Justifed or not. ln some ofthese cas there ks no other theory, and they cannot be just îgnored.

Besides the domain wall dîscussed in chapter 8, one-dimensional modelshave been used in a.n attempt to find the eS'e'I;t of planar crystn.lllne defectson nucleation and coercivity. The physicat properties in a certain regionwere assumed to be diferent 9om those in the rest of the material, andBrown's equations were solved separately in the perfect and ia the imperfectregions, and then matched together. This problem was irst solved (47% fora defective re#on in which only the anisotropy constant waz diferent fzomthat in the bulk, aûd later extended (478) to a modifcation of the exchangeconstant C and the saturation magnetization, Ms, besidœ the anisotropyconstant Xz. The former was later used (4794 to explain the coercivity of ahard Yateriat. The sam6 model waz revived for the case of a wall which isassumed to be already in the defective region, iastead of its being nucleated

218

there. The coerdvity in this case is det,ermimM by the pinning of that wallto the defect) which i.s mathematielmy the s=e problem ms in the d-lFerentphysical caqe of t'he nucleation. Retzlts were reported for a region fn whichvth C aud A'z were mod-t6ed 2480) 4811, and for a case in whicâ only Cwas chauged, as diFerent fnnctions (4821 of spMe. There is also a case ofa fzlm (4832 with a =iable thiclmeœ, for which the mathematiœ is stillessentially the same.

With the new interest in multëple fllmK, the samz onedimensionalmodel has been llm.M for studyizg strongly coupled ftlms. In tb.is case thereare ae two regioms (which are the flms of dlFerent compositions) wkththephyfcal const=ts C, Ms and Kz being dlFerent for each of the regions.'Because of the coupling, the solution in one region should pass smoothly tothat in the oier region, so that the mathematical problem is ldentical tothat of (478), although these models are always rdnvented without pa' yingattention to tîe previous work. Rmalts ha've been reported for two flms,enzt% of which is œsentially saturated, vdth only a transition layer betweenthem bdng a function of spRe (484), an.d for a cmse of 1111 variation overeRh of the two ftlms (485J, ms well as s'uch a variaïon for two 'Flms with anantiferromagnetic (486) coupling. Detils will not l>e #ven here for any oftkese cases, and it will only l:e rqrnnmked that the mssumption of one dsmen-sion may be too restrictive for describing the physical situadon in manyof the problems to whie s'ue,h models are applied. Evem a large Xl173 endssanewhe% aad it %nA been noted (48% that the efects of the edges maysometimes be very large and vnn invalidate the one-dimensîonat approach.

An mvxmpleis shown schematicvy in Fig. 10-1- In (a) the ma>etizationis parallel to the 11lnl plane, m'eating a charge on the surfve. 'WhG thebard material tA' is magaetized to the right, the Geld due to its surfacecharge points to the left, and the SOf't material tB' e>n be in a negativeEeld when the applied feld is pMtive. In (b) the flms a2e not continuousand one matarial Tpenetrates' through the other one (which tun happenin practice). The applied feld and the magneœation are perpendicularto tàe flm, but the surfaœ cbarge of 1A' can st111 crea* a feld at $B'in tlœ opposite dizection to that of the applied fdd. Such cases can #vea magaetizptinn cm've whic.h is qualitatively diferent (487) from the onecalculated by neglecting thœe efecks- Surhce roughness exn also (4884 causea m'mlnar demaoeœation.

Nevertheless, suc.h calm:lmtions do ezst in which this usumptsion of onedimeasionality Ls made. Md once it is made, one may ms well A-xlre advaa-tage of a particular pzoperty of one-dimendonal micromagnedcs which isonly little known, although it caa fadlstate the res't of the calculation ve,zycondderably. Before spe ' * this theorem, it should be noted that imatrue one-dimeaslonal cmse, Le. when M and U are functions of only (say)z, eqn (6.1-4) becomes .

ANMZYTIC OCROMAGXTICS

FHBT MVTEGRAL 219

+A ---) +

+

B

+A ---A +

+

- yI -

---TA I y(a) (b)

FIG. 10.1. The edge of a multjlayer f)m made of a hard materhl tA> anda xft material CB'. Surface charge is shown schematicuy for the layerszmvnetized parallel (a) or xrpendicular (b) to the <lm plane. Copiedfrom (483.

(f2 V . !.y .-z.d m= 4xMs , (10.2.3)

wlticlt is zeadily integrated to

(10.2.4)

The constant of integration depeads on the gometrs but it eAn always l)eabsorbed into the demagnetidng fRtor. The whole term in eqn (10-2.4),whem substituted ia eqn (8.3.34) and then used in Brown's equations(8.3.37)-(8.3.39), has the form of a unim-al anisokopy term. Therefore,the wùole maaetostatic 0e.1.& may l)e left otlt fn c *1%6 drsd-dfmerlsiorlclcdcmlaiion, aud hduded only as a modifcation of tke anisotropy. Tàis fea-tnre did not appear in the domin walls in thin flms discussed in chapter

. 8, bœause in that cxqe the calculation was not a true one-dimensional one,with the magnetostatic enerr of the surface azong 2/ superimposed oa theassumption of no dependence on y. However, here the term of eqn (10.2.4)will be written senrately, and not induded in the azisotropy, in order to

Au= 4rKmx.

220 ANMYTIC MICROMAGNETICS

emphasize its eisteace-Substltuting in eqns (8.3.37)-(8.3.38), fœ the c%e whea m does not

depead on y or z,

aelmz rng tAm,z .s. .xxjssmz - l'nz

uyC' - .

; + Mk xdx9 rrsa dz ru

ov= zn.z ê'tsw- + - .. = 0,

:mz mz Onzz (10.2-5)

aad

fvnu m...x g2ru m--y.g Aa rN. 0m.c - + us Jz; - a - + . = q*2 mz dmz mz êm, mz dmx(10.2.6)

where H now contains at most some demagnetizlng fartors besidœ theapplied feld Eo, and ms such îs a corl-dcnt which do% not depend on z.

Consider now the erprexqion

a a a1 d'mn gwzx dmz va z.A = -.C + + + M - H - Ya - 2r s m..2 #.z dz dz

(10.2.7)lf this exwession is Werentiated MY,N resped to z, one should use

doa, 0m= dmz o'tru dmv d'tpw dmz= + + j (10.2-8)e 0mz dz 0mv ti'r dmz (â

because 'ttu depMds on :r only via t:e components of m. Substitating forthe second dezivativœ of rrzx and mv 1om <ns (10.2.5) and (10.2.6), it isseen that the coeëdent of d2ma/*2 is proportional to

tlm. drrsg dmz 1 d amz- + my - + rr;x - - = - - (zn ),(2.,5 tf's dz 2 lLc'

whic.h is zero because m2 = 1. Also, the sam,e factor multiplies H. andt'llpa/gmx, aad on the whole it is seen tîat

= 0, .A = coastant, (10.2-9)

for any function m which fuïls eqns (10.2.5) and (10.2.6). In other worcks,X is a Srst ïnfcprc! of Brown's cquations ixk one dimensioa.

nin expression wms 6mt proved (489) to be frst integral for the par-tirtnlnm case of a Mm-aial xniqotropy, and then generyzzed (270) fœ a-nyanisotropy. lt %nA not bee,n generalized to more tltp.n one dimension: but

BOUNDARY CONDITIONS 221

it applies as wdtte.n here to cll publjshed onc-dimensional models of thediferent physical problems, with constant C, ffz and Mst fncltldïn.ç thecases of these eonsunts ehaaging abruptly fxom one <ue to another in

h diserent regious. It does not apply only to the cases i.zt whtc.ll one or moreof these constaztt ch=ges continuously over a (mlxt'n re#on of spaœ, sucàas in (477) aad (4821, althougk it should not be diEcult to genprnll-vo itfor these cases as well. H some of the published case this frst ixteralhas ben rediscovered for the particalar case under study. H others, it 11%been ignored, evea though its use could have simplifed the calculations.h particular, when the dilezenfsnl equations are solved numerically, it iseuier and more accurate to solve a Erst-order equation than a second-orderone, and it kelps if one of the constants of ixteration can be detcmineddirectly fxom the physical problem, instead of having to be adjusted durixgthe computations. Sometimes the use of this frst integral may even lead(490) to a complete analytic solution, mnxking the numezical computadonsunnececx4nry.

Theumefrst integral has also been used (334: 491) as a self-consistencytest in the computations of certnx'n one-dimensional domain walls. For tkeremanent state, H = 0, another frst integral was also found (4894 forthe cmse of a uniax-ial anîsotropy. Together witk the present A it allowsa complete analytic solution of al1 one-dimensional problmmK in zero feldaad this n,nssotropy. It has not beea gowmlie,p.d to any oGer cx-.

10.3 Boudary CondltlonsMatrlu-ng together the soluhons i.a dfereent re#ons, as mentioned in theprevious section, usuallymelmthat both the magnetization and its normaldeziwïve are continuous on the surfve between the two 1:/:)% of materials.TMS condiûon is quite obvious where Ms is the same on b0th sides of thatsurface, bacause the exiange is very strong over short range (see section6.2.2) and this energy te= prefers neighbouring spîns to be parallelto eachothcr.

When Ms is not the same for the two materiah the same purposeis served if both the d-ction of the magnetization vector and its no=a!derivadve aze contiauous on tàe boundau. h one-dimensional calculations,ms in the predous section, this requirement means that the angle betweenthe mMnetization and the z-ads is conthuous and sm00th. This conditionkas been used in almos.t a11 the calculations of th1 sort, ucept for (4&S)in which an angle dlcontinuity waa usumed, and was deâned i.zt termsof a certaân unknown prameter measadng the exftbx-nge coupEng at theinterface. This approacâ was criticized in (48% as bdng too drutic a cxaage-It may, however) be necesary to taV into account the possiblty that theevhxnge coupling at the ixtprlnzne between two materials, being diferent1om tYat witlkin enzth of the materials, may add an extra term to theboundazy condition there.

222

The coneuity problemis not limlted to t%in 61nu.R is also enountaredi.n othe,r tyws of heterogeneous materixh, such as magnetic alloys in wàichlYons of dieewmt chemical coppositions of the alloy may edst (492, 49%494) in the same sample.Another exnmple is the so-called tcobalt-modiied'p-FeaOa, ia which particl% of 1%1K ferric oxide are coated by alayer of cobaltf-'te. The theory in this c.ase (34% 350) Jtlqn %stzmed a consinuity of themagneœadon drection and its derivative on the boundary between theey-Feaoa aad the CoFezozl. Agaizb it d- not prove that otàer boundocondktions should not be used.

One pcssible way to modify the boundary conditions betwœn suc.h twomaterials (492) is to postulate a Mnd of surfn- integral, of the geaeralfozm of an exchange htegrall wlziclz is supplsed to manifœt the d'-eerentexn%xnge on that snrfn ce. For the case of no oth.er surfaze anisotropy tum,s'tzch a postulate leads to the boundary conditions

ANALYTIC YCROMAGNETICS

Cï 3MIMs x - - rlaM: x Ma = g; (10-3.10)

(10.3.11)on t;e surhce which xllarates Mz (with exchange consunt %) Md Ma(witk exchaage consunt &z). Here nz and r?,a are the normals from dtherside of the interhce, and Jt'zz ks a Nrameter of tlze theorp

These boundary condttions have the a4IVa,IZta,D that they reduce to theconventional ones of eqn (8.3.42) for the Bnn't of a boundo between aferromagnet and a non-fezwmagnet (Ma = 0), in the abseace of a stzr-fm anixkopy. They have the dindvantage (2702 that they do not reduceto some tzi.vùal continuity requirement in the limit Mz = Mz when thebounde is jnst an arbitrazy mlrfnzte insidû the ferromagnet. In this b'mitMt x Mz = 0a and eqn (10.3-10) or (10.3.11) leads to M x 0V(0n = 0,which is an impossible requiremeut for every arbitrau surfaœ inside theferromaaet. lt Ls non-physical to have a special form for cnly the boundarysurface, witiclz mnNs tt tmposstble to adopt these boundary conditions. Thephysical problem, however, Ls sttll therw and more appropriate botmdaryconditions may still have to be developed.

10.4 WaII MassThe motion of a real wall through an implrfnct material is quite compli-cated, aad outside the scope of œs book. Discussion is limited here tothe case otan ideal, stminht wall (xxnlslce tàe bubble wall, wilich is (495)a dieerent problem), moeg in a perfect crystal with a pedectly smoothsurxe. Ev% in this case, the wall structure cannot be the same as that of

. a svlonary wall, because of two r-ons. One is the Xect of the applied

52 ôlvzMc x - .R';aMc x ML = 07x'kf'z drpa2

223

feld whiG drives the watl, aztd the other is the gyrouuxnetic eEect, aseeressed by the dynamic muation in sectioa 8.5. Only the second one willbe described here; and only for an undamped, uniform motion.' . All early wozk on this pzoblem) suc,h as (4962, started from the specifc'%sumption of a one-dsmemsioaal wall confguration. Even iu the work (49%whicN could be readily exteuded to thre dimensions, acd in its extensions(498, 499, 500), the actual cxample were those of a oae-dimensional .wa11. lnthese works it was noted that 7t ofeqa (8-5.48) wms ae thevariatîon oftlje-enerr deRity, w, which should also be quite dea.r fzom the dirivation'of Bzown's equations in sectioa 8.3. Therdbre, if M is replaced by izrlirection, wbick nn.n be upressed by its poiar and n.ezimuthal angleq 0 and4, eqn (8.5.48) is actually

do Jj.JYsin#uu - andW M s %-$(j'. c'p 'yn J'tn-2- sin 9 = ' '

,d t uk' Js tj- 8 (10-4.12)' where J designates the variational derivati've. ln the particular cx* of auniform motion at a velodty 'n izï the z-directioa, the derivative with zespectto the time may be expressed as a derivatlve with respect to z, according'j;(:)

(1 d8 ds d4- = -v- and = -v-. (10.4.13)dt da (b- dà

In this particular case it is seen (501) Oat eqn (8.5.48) exn be rewrittem' é' j'

Lm - '?rzl =

y (0 - '?rzl = 0, (10.4.14).wheze

Ms% #01 = cosd-.'p dz (10.4.15)

.' This result mpAns that the dynamics of a uniform motion can be taken into:: azcrqat bar minimîclag the iatagral of 'w - 'tt)z instead of the Ecuir-izr.tzatiouQf the integral of w in the static exsf!.

Consider splvn'6cally the case of a moeg domain wall, which has thesHe statics ac in iapter 8, and with the same geometry ms' defned by

' Fig. 8.1. It witl nlM be a-qmlm ed that tàe wall structure doœ not dejend6n z. ln tln's cmsea the foregoing conclusion means that the enera per unit*a2 area of a uniformly moving wall can be txt'nn as that of the s'tatioaac*al1, plus a dynamic f,cr)): (501),

WALL MASS

b xlem = -- mjdzdy,

25 -& -x(10.4.16)

where 'u?z is defned in eqn (10.4.15). Chan#ng fzom 4 and # to the moreconventional Cadesian coordinatœ of m, this dynamic enera term e-qn be

$ written as

224 ANALYTIC MICROMAGNETICS

M % b * m drzz omu% - -c5,'â J, /-- mz

+-7'e, mv dz - m. dz dzd''. (10.4.':)

SucN a mim-rnimzttion of the integral of %v -'d)z îs equivalent to minimhingthe Izagrange function, dzlnGl as the pot'ential enerr znf'n'?zs tke Mneticenergy, i.a mechamics. Therefore, it is convenieat to defne a 'tllall rpad: mwau,so that the ldnetic enerc is equal to 1a-q*2, an.d write this wall mass

per unit wall Jkena as2J0

'rawall = :) , (10.4.1S)after rninimw' img the Lagrange fuetion. This expresdon is no$ nec-qm-lyindependent of t), and the mmss may depend on the velodty. It is oftenfound (502, 503) that the bAbxviour Tsnn be approfmated by

T4D0<1I =

1 - @/,?7x)2 (10.4.19)

at least for rather 1ow velocities. The mass zn,o in the llrnit % -.+ 0 is knowna% the Döring mass, after Döring who had predictM the efstence of suGa mass alr<mz!y in 1948. It should be emphasized that the wal! maœ in thinfzlms Ls a real entity, ud experiments on wall motion indeed skow (504) abehaviour similar to that of a particle with aa inerkial mnAs-

When written in. the form of eqn (10.4-17) it is dpnr (501j that thiskinetic eneràr, aud therefore also the mass, is identicclly ze- for all one-dimensional dozal wall models of secdon 8.1, becaase either mz or mvis identically zero in a1l of them, which mxkp-q the integrand alwazs zero.Many workers managed to obtain non-zero values of mass fzom these one-dimensional wall modelsj but it vzms only because they did rt/t use ecn(10-4.17). In particular, Sehlömann (499) noted that eqn (10.4.15) was also'another posdble choiœ' for ms, which he preferred to wzite in a diferentform. There are, of courset other forms in wlkieh to writm the Lavaagefunction, but the mint is that they should all lead to the same wall structureand energs after proper mimlmlqation. If they do not lead to tke samcreult, one should not just choose among them, but realize that the diferentresults meazt that it is not an enera minimum. The logic is the ume asin the self-consistency cheaks in section 8.4, and indœd it should be clearfxom cxapter 8 that even the stadc onodimensîonal wall is no'l a properapprozmation for a m'-nimal enera wall stracture in 61ms. Still, a recentlook iato some relations between the digereafsnl ecuations (505) gces backto the picture of an esseatxy one-dimensional wall.

It cau also be seea from Kn (10-4.17) that the ldnetic enerr and themass will both m:n-'K% whenever mx is a.a odd function of y, while rzvand mz are even functions of y. rrnu'n symmeky is found izk a11 tàe two-dimensional stationary walls in zero applied Geld, and it mpst therefore

TII-R MMANENT STATE 225

be concluded that an additional asymmetry in y is added when the watlmoves. Such aa asymmetzy is indeed found in the study of Hubert (249)()f a domain wall h4 a non-zero feld. It ks aISO found in the computations

. of movîng wall structures which will be desHbed in the nex-t chapter. Fœllniq reMon, models of moving walls have been constructH (50% 5œq withthis Mnd of azymmetry in the A-direcdon.

10.5 The Remanent StateAn argumc'ai presented in section 6.2 wms meant to convince the rtmoder thatthe total enera of sulciently large partides is reduced by subdivision intodomains in zero applied îeldy whi)e for a small prtide the exnhuge is toostrong to allow it, and the particle should remain a uniformly magnetizedCsingle domnln'. The study of mtpepramagnetlm in section 5.2 is basedon an even stronger assumption, that suEciently small pmicles are alwaysuniformly magneœed. This x'anmption is actually a little ioo strong forthis pumose, beeause superpar etism is not always due to a coherentrotation of the mMnctization. It has been demonstzated (180) that undercertain circumstances, the thermal Quduations caa excite a back<d-forthmaaetization revlmztl by the nurling mode.

This (jnxb-vtive ar/ment eAn be made quantitative, giving the sizeuade wbzc,h a pn.riicle xs a single domain. Sometimes the procedure ts tocompre the energy of the uniformly magnetized state with that of somecllcfen spatial co rations of the magnetization, as done ia e-g. (283, 3464.But as in tie cmse of Brown's equations for the maRetization process, thiscomparison is not suEcieat: because there is always the riRlc of overlookmga state whose energy is still lower. Thus, for Rmmple, the fk'rs't modernstudy of this problem in a spheze compared the (252) Gergiœ of all thecoaf,gurations whic,h could be obtained by c'utting the spheze into slicc.But it then tumed out (2534 that a still lower eaergy eltn be obtained bydiddng the sphere into cylindrici domainsj whic.h cxnnot l:e expressedby that sEcing. lt is Gerefore necessary to tzklr- into account all pceblemaaetizaton confgurationsj and to do R by a rigoxous calcttlaton withno approvlmations. TMs problem, whick Brown nxme %he fundamentaltheorem', can be stated (507) as claiming that the state of lowet eneraof a ferromagaetic particle, whose size is less tkan a certain critical size, isone of uniform magnetization. It also goes v'ithout sayins thai above this(xitical sizey the lowœt-enea state is one of non-uniform magnetization,although it doe not necvarily have to be a subdivision into domains.There eltn be states of non-uniform magnetization just above the criticisizej which are not the fully developed domain confgurations, with thelatter coming ia only at a still larger size. The proof of this theorem willbe #ven here for the Mmple geometry of a sphere, before discussing othergeometrie fœ whicx the problem is still controversial.

226

10.5.1 SpltereThe total energ.y of the dîFerent coMgurations considered here is the slmeas in eqn (8.3.22), except for Cu which is omitted herej because the presentYculation is for H. = 0. The integrations are over a sphere whose radiusi.s .2. lt is also assumed here, as in the calculations of Brown (507, 508), thatthe surface auisotropy ks zero. Experimentally, this anisotropy Ls not zeroin many cmses, such as g2OO, 201) 202, 475) (see also end of section 11.2).

H the single-domain state; namely when the sphere is uniformly mag-netized in a direction parallel to an easy a'ds of the anisotropy energy;éle = éla = 0, and the total enerr is the magnetostatic term. The latterhas already been calculated for a uniformly magnetized sphere in (6.1.14).Therefore, for this state,

ANALYTIC YCROMAGNETICS

8x2î?7 aeorm = 'V =' X3MZ.u 9 s (10-5.20)

The energy of a11 the other possible states has to be compared with thisexpression.

For the energy of those other states, lower and upper bounds for Cuare calculated, according to the generk idea of the technique described insection 7.3.4, although the particular trick used here for the lower boundis not mentioned there. For fndiug a lower bound, the constraint of eqn(7.1.7) Ls replaced by the meaker constraint,

a z 4*(mz2 + mv + mz) d.p = zc = y+t (10.5.21)

where the integration is ove,r the sphere, and 't; is the volume of the sphere.This constraint allows funcdons of space which are not allowed by thestronger constraint of eqn (7.1.7). lt mmnns that the search for a minimumis done in a larger group. Therefore, a minimum found for the weakerconstrxint may lx due to a fanction which dces not belong to the original

' group, in which case this minimum is lomer than the lowest minimum inthe ori#nal group. It cnnnot be higher, because the weaker constrint alsocovers al1 the fanctions which are allowed by the stronger one.

For calculating the lower bound, the anisotropy eneza is also omitted.Whether cubic or uniax-ial, the anisotropy energy term is always positiee,and it is legitimate to omit a positive enera term for calculating a lowerbound, because such omission decre%es khe total energy. Eneror rn-pni-mization under constraint is carried out by the standard use of Lagrrgianmultipliers, leading to the three dsferential equations

47r zg&V2 + zj ma = Mx (m(x), for a = z, y, or zt (10.5.22)3

where l is a constant Lagrangian multiplier, and

THB REMANENT STATE

1..17n t ;m . w Jg'h y / .,!y ,' r

227

(1.0-5-231?1

(znzz) = - mzdv,k?

The boundary conditions are

(10.5.24)'. The constant à and any other integration constants must be adjusted sokthat the constraint in eqn (10.5.21) is satisfed.

Multiplying the equation with a = z of eqn (10.5.22) by m.t the onewith a = y by mv, and the one with a = z by mz , adding, integrating4 ver the sphere: and using the divergence theorem and eqns (10.5-21) and(10.5.24), the enera of eack of the solutions of these equations can bebritten as

1fkoa-zaaiorm =

y, l'llh (10.5.25)see also section 8.4. Comparing with eqn (10.5.20), the t'atio of tkeenergy ofany of the solutions of these dWerential equations to that of the uniformlymagnetized state is

3,LJ?- = - . (10.5.26)4$,r xl.f Zs

htegrating now 'a-qztb of the muations in (10.5.22) over the sphe're, andusing eqn (10.5.24) and the defnitions in eqn (10.5.23))

0m% :mv 0mz. = = ... = 0) onar dr Jr

' l - 4/Ms2) Lmn; = (A - 4-/ Ms2) (mv) = ,k - 4-J Ms2) (zrzz) = 0.T(10.5.27)

1.î tzn.zl # 0 or LmvL # 0 or (zrzz) # 0, this equation means that

4/ a.ï = wM, , (10.5.28)

which according to eqn (10.5.26) means that the energy of such states isqnal to that of the uniformly magnetized state. It rltn thus be concludedthat the energy can be smaller than that of the uniformly magnetized stateonly if

(Tn'm) = (Tn'?/) = (Tn'a) = 0- (10.5.29)With the substitution from eqn (10.5.29), the equations in (10.5.22)

are linear and homogeneous diferential equations, whose solution is wellknown. One cztn write, for exwple, for mz the most genezal regular solutionin the form

m.z = Ain.iknbur/RtkkgLo, /), (10.5.30)where .â is a constant which has to be adjusted to satisfy eqn (10-5-21), j.xis a spherical Bessel f anctions, Fs,s is a spherical harmonic, and knyu is the

228 ANALYTTC MICROMAGNWICS

rth solution of dJa(z)/dz = 0. The latter condltion is necessary to satisfyeqn (10.5.24), aad it is san that all the equations wkich involve mz azefulftlled, with A of eqn (10.5.22) being cxqunl to one of the dgenvaluu

ce2 = RIM

%9# yg ) (10-5.31)

for atl allowed valuu n and v. Hbwever, according to eqn (10.5.25), theenergy incrpzmes with A, and for the lowest-eaerr mlm-rn =vn the smallestl should be iaken, which is

'

(7q2,). = 2 (10.5.32)/2 '

where h = 2.0816 is the swmlledf root of eqn (9.2.27). Note that for thissolution the other component must be mz = mv = 0, because any othexsolution of eqn (10.5.22) is not compatible with the snme 1. A slmiln.rsolution 5s possible for mz or rzv instead of mz, but its eaira is the sae.Tkerefore, the smallest energy for alt possible ftmctions whic.h solve ikjslower-bound problem is given by mn (10.5.32).

Subsdtuting in eq= (10.5.26), it is seen that R > 1: namely the lowestenera is that of the nnifovmly magnetized state, if

(10.5.33)

lt has thus bvn proved that the lowest-enerr state is one of a uniformmagnethation for a sphere whose radius R fnl6ls

cqk 47 a> Ms -YY

.tc- /-3c l . 0 17OR < Rc0 = j/ k-é- A$ u

- -

Ms s(10.5.34)

lt should be noted that, even without an upper bound, tMs rauit alreadyprove the statememt that the nniformly magnetized state is the one whkbhas the lowet eaergy', for 'sttEdently small' spheres. For a quantitativeevaluadon of bow smil SshlfBciently small' is, aal upper bound is also needM.

For Weulating an upper bound, it is Erst xtoted (508) that for aay realnumbers mx, zrla aad mal whose xparœ add up t,o 1) '

2m2 + m2m2 +yn,2m2 = (m2 + m2) - (m4 + 0,20,2 + a4) < ,.2 + 0,2T'V y v z z z z u x :r v p z u.

(10.5.35)Therefore, a cubic anlotropy energy is always Mcller than a txrn-azalaaisotropy with the same fft . Since it is always ie#timate to ïncrwlae thetotal enerpr in calculaiing a'n upper bound, any upper bound whicà iscalcnlxted for a um-av-al anisotrops of the form 'tsa = Jfztmzx + m2), is alsovvalid for the (1,9e of a cubic aisotropy with ihe >me value of A%.

THE REMANENTSTATE 229

The teclmique for Yculating an upper bound in Gis case is bazedon resGcting the enera m'-nlma-em.tion to a palticular class of 'hmctions, oreven to one particular function. Suc: a restriction may dalirnn'n ate the spatialvariation for whiclz the energy is the lowest mknimum, so that ihe lowestenergy of the special restricted class is larger than the real minimum. 1tcrnot be smaller, and nn.n ai most l)e equal to the real minimmm, whenthe lowest-enera state happens to be iacluded in the retricted nln-qs. Inihis respect, a computation such as in (2521, whic.h rnI'=,'m1+zx the eneraof al1 magnetization confgurations that can be obtained by slidng thesphere, is also azt uppex bound, because it considers a pardcular class offunctions. For the preseat problem, Brown (507, 508) considered two ldndsof functions- One is a rough imitadon of the curling mode,

m6 = 1 - m,2, (10.5.36)

where p, 4, and z are the eplinddca.l coordénates. The second one is a roughimitadon of a two-domain structure, with a wall between ihem, taken as

mv = 0, for

m. = sin (Ps ) , mv = cos (Ps) , for - ?z < z < h, (10-5.37)mv = 0) for z > h

where h, is a paramete with respect to whic.h the enera is minimized.An upper bound to the enerr can be calculated analytically (5084 for

tozw.ll of these particnlv fhnctions. Afœ Kmparing this energy with thatof eqn (10.5.20), the result is that the lowest-enerr state is that of anon-uniform maretization if R > Scz , when eqn (19.5.36) is used, or ifR > Sez, when eqn (10.5-37) is used, where

2prtv = 0, ma = 1 - - ,R

4 .52 927-CSc, = ,.v/4xM: - 5.6150A% (10.5.38)

provided that ihe expression under the square root is positive, and

9&.a = -

a (T'lft% + 8wvM' ;) , fr = 0.785398. (10.5.39)8 t3t>' - 2)M,Since two separate functions are nsed, the smaller of the two radii may beused, and it is possible to state that a suEcient condition for a nonluniformmagnetization state to have the lowést enerr is that R > min (Sc$ j Sez) .

Equation (10.5.38) is useful only for small 'values of A'z-lt is meaniaglessaud invalid if JG Ls so large that the exprt-on under the square root

230

becomes negadve.Bven when this expression îs still positivej but small, thisequation is not usdul because it leads to a very large Rcz . Therefore, eqn(10.5.38) is applicable only to soft materialsl with 'mM.1 << Jf'zl leaving onlyeqn (10.5.39) for larger mlues of Kï. In practice it tmrns out that the latteris not very useful either for very large vaâues of Jf'z, and c,an lead to an Rcgwhich is orders of mn.gnitude larger than Roo. Knowing that the turaoverfrom a uniform to non-unif orm state ks somewhere between Rcz aad Re: isas good as knowizg thks transition to within 10% or so for ver.g small JG,because thee radii are that close together. The situation is quite dxerent, .however, for very large values of Jf'l, for which the preent calculationleaves an uncertakty ia the order of mn.gnitude, obviously because thelowe,r bound Rec of eqn (10.5.34) is too small. A bound whic,h neglects Kzcannot be expected to be close to the correct Nalue when Xz is large.

There are some indications that for soft materials the lower bound isclose to the e'xact valuej while b0th expressions for the upper bound leadto too large bounds, aad should be replaced by better one. In certaincomputations, to be described in section 11-3.2, on one caase of a uniafalanisotropy (253) and two cases (5091 of a cubic anisotropy: the computed val-ues were much closer to the lower bound than to the upper bound of Brown.In prindple, the results of these computations are only upper bounds, be-cause they apply a certainconstraint, and do not really mlnlmlze the enerrof al1 possible functions. They can only be presented as good approfma-tions to the tzue, three-dHensional structure àccc'lzsc the upper boundsthey lead to are very close to the lower bound of Brow'n- There are azsoexperimental data obtaked (510) 1om neutron depohrization which showa dear traasition from one to two domains in Mno.eZn0.asF'ea.0sO4- Thistransition is quite close to the lower bound of eqn (10.5.34), and very con-siderably below the upper bound. Of course) the particles in this experimenthad a rather irregular shape (510), and were certainly not spheres.

There is also another analytic upper bound, eve,n though it has notbeen presented as such. It started from an attempt to approzmate theconfguration of the curling mode in a sphere after the nucleation stage,but was then actuaâly used (462) to fnd the remanent state, and for someestimations (2.80) of superparamagnetism by curling- It ia an upper boundfor the enerpr of the remanent state, as is any calculation which resthctsthe rniniml-so.tion of the enerr to any class of functions. H this particularcase, the assumptions are (1804

m, = 0, m.z = gzLrb + (1 - gc(r)) cos2 0: m4 = 1 - m2z, (10.5.40)

where gz is a function of the radial spherical coordinate, with respect. towhich the enera is mlnlmlzed, botmd by the constraint #0(0) = 1 to avoiddisculties at the centre. Wlze.,n this assumption is substituted in the expres-sions for the enerr, a diferential equation can be written (462) for Jo(m)

ANALYTIC YCROMAGNETICS

T% REMANENT STATE 23l

whicb mlnivnsmœ the e-xchange, anisotropy and magnetostatic energies.This dlFerential equation hms tb be solved numerically, which is a much

éasier task th= a numerical solution of the whole problem: because onlya one-dimenïonal, ordinazy d-eerential equation is tvolved. Still, it is notas eas)r to ux ms the foregoing result of Brown, which can l:e expressedin a closed form. Computation has only been carried out for the case ofl,1n1>='al cobalt, for whiG the critical radius was oaly slightly larger thaak.hat computed (253) for all p-ible dependence on r and 0. The functionalto= computed for this model was al= quite im''lxr to that computed in(253). It is made of two e#lïnddeallp symmceic domrdns, mavetized incopposite directions, and separated by a spedal Mnd of wall.

10.5.2 Prolate s'gher/ïd' .denersllrdng Browa's lower bound to a prolate spheroid (511) is quitessrairyhtforward. A1l it actually txlrp-q is to use spheroidal wave Ganctions, in',spheroidal coordinatesj instead of the spherical onœ nsed in the previousrsedion. The caze considered here is a prolate spheroid for which the easyr ,;t='E of the itnlqotropy % pazallel to the long n.=-K of the spheroid, wlticà is't,alte,n as the z-ais- The semi-xvis of tàis s'pheroid atong c; or y is denoted'b.y R, aad the demagnetMng faztor along z twhich replacœ 4r/3 of the'sphere) is deaoted by Nm. The result Ls (511) that the ldwœt-energy statefor sucA a spheroid, in zero applied âeld, is one of a tmiform 'maaetization,èhenever

q CR < .&: = .- , (10.5.4:)Ms JfzFhere q Ls the pnmmeter deâned in section 9.2.2.3 foz aucleation by theEurling mode in a prolate spheroid, with an analytic approfmation in eqn(9.2.32). Actually, eqn (10.5.41) Ls identical to eqn (9.2-33), except for Nzh hic.h has bœn replavd b,y Nz here. This diference shoald be obvi-t ere w

oms, b-.use the non-uniform magnetization state heze is compared witk aunifnrm magneuzation along the z-directionj whereas the cuzEng mode insedon 9-2.2.3 is compare with a coherent rotation izt the z-direction.

It should be emphasl-mzw.d that this resalt applies to the remanent stateonly. If a partide is uniformly maaetized in zero applied Geld, it doc notnecessarily folbw that it wi.ll remnsn so whe.n a âeld is applied, and it doesùùt necessadly follow that it will reverse its magnetization by the mherentrotation mode. In spite of the s5m-11A.r1 ty in the mathematici exprpnqions fort.

, the tcritical size' for chaaging ove,r from one state to Mother, or fxom onemode to another, remanent states aud magnetization reversal modes aredieerent physicaz problems-ln particmlar, the lowut-eaera statein a #ve,n

. field may not even l:e reached during cfarfrm stages of tbe magneœationreversal process: as explained in Gapter 9. It is quil posdble, aud pvnm plœ

' were given there, that a lower-enera state ezsts, but the system cannotreach it because of an energy barriez in betwœn, aud it gets 4stuck' in a

232

higher-enera one. Besides, the diference betw-n Nx and Ny exn be vecsignifcant fcr elcngated elpscids.

A plot of CJV.W Z-Z'M be found in (5114, wlhic,h should give an idea ofthe values iavolved for not-to-longated ellipsoids. For = aspect ratio ofaboqt 3 or larger, that plot Ls not aec-aty, because qï! Nz becomes(51:) a remsonably good approfmation for qlvm, where t)z = 1.8412 isthe limiting value of q for an '-n4n-'te c'ylizder, as dezned in section 9.2.2.:.

Aa qpper boqnd has not been calculated, nor is it necessary for îndingout what the sitqation is for typical partida used in practical recordiagmaterixlK. Thus, for evxmpleo qj x ra 3.1 for an aspect ratio of 8:!, whichyields a lcriticap dixmeir of at lpxAt 2./% ;4$ 150 nm, for magnetite withg511) C = 1.34 x 10-6 erg/cm aad Ms = 480 emu. Other e-xltrnplœ areaISO given in (511), vdtk the conclusion that cll particles n!o.d in rrrdingmedia n.'> segle domnln' in zero applied feld, in as much as theîr shapemay be approzmated by that of an ellipsoid. A more recent maten'rtl forperpendicular recordhg (512j is made of aa array of parallel nicltel px-lb.rs,with a uoifnrm diamete.r and dîem:n ce. Using for nîckel Ms = 484emu andC = 2 x 10-6 erg/c, for particlœ with a distmeter of 35 nm and a heightof 120 nm, eqn (10.5.41) yields a critical radius of about 50nm , so thatthere rltn be ao doubt that these parlcles are uniformly magneeed inz&o appliv Geld- Thks estimate is Yready saëcient to make sure ihat theothe,r sample (512) with a dinae*r of 75 nm is also made out of singledomains, which would have 1*e.n trqe ev% if ît were the spxrne apect ratio,and is even morg sn vith the larger aspect ratio for this spxrnple.

Foran ellipsoid whicx is not ver.g diferent 1om a sphere: there is also anetpansion (513J arotmd the lower bound for a sphere. This eNmndon is notnecessav for ellipsoîdc, now that a rigoroms solution is Hown for prolatespheroîds, aud it should not be diEcult to extend to oblate spheroids aswell. It may be po%ible, howev&, to adopt such aa expaasion for shapeswhch are not ezipsoidal at all, as is the case in reeal pardclo.

10-5.3 Cnbe

ANALYTIC MICROMAGNEWCS

Most computations on the remanent state of a: cube t=d other shapes),in the size range which is normally dm4nu to be a Sfzie particle', wlll bediscussed in the next Gapter. For the pmicular case of a cube, there is aseeMngly complete study (514) of all possiblefunctioas of space, which cmn

be expressed by cutthg the cube into slices. TMS computation is rigorouswithsn iî own lamework, but the constraint of a one-dimenional slicingmakes tàe critical size tàus obtained only an qppe,r bound.

There is also another problem with sue.h computations for a cube, oractually for any non-ellipsoidal body. The d-agnetizing feld inside suchbodiœ is not homogenmus when they are uniformly magnetized, and itseems that the Eeld components Nrpendicular to the magnetization vectorYust eaforce some non-nnif ormit'y. Moreover, for a unilormly magnethed

THE REMAMNT SVATE =

cube (or a fnite c'ylinda, or any other body which has a sharp corner)the demagne - - Eeld is formally insnite at the cozners. The integratedmngneuutic enera is Enite, but some (431, 515) nlnlrn that it is wrongto use this Enite enera as in (514J, because the local infnite Ecld wlllnever allow a uniformly magnettzed confguration. This clztim has beenœntroversial f or many years, because it cltn also be said (3921 that a sharpcorner is only an approzmation which cannot est on an atomic scale, auymore than a smooth, empsoidal surface c-an (see also section 11.3.S).

The calculadon described in tMs sedion is restricted to an unusual caseof a cube made of a small number of atoms, in an attempt to undezstaadthe transition to the atomic limit, by patticularly avoiding the use of theapprovimation of a conttuous material as htroduced in chapter 7. It hasalready been mentioned in that chapter that this approvlmation musi brpltlcdo= if the theory tries to de.al with dis-taacœ of the order of a unit cell,and it is always possible that even tens of unit e.qllK may be too small forusing safely the app-vimations of micromagnetics. It may be necessazy tostart wondpn-ng about using this thezy for casœ where the whole size ofthe pardcle is oaly a few tens of unit cells? as are some of the vene smallpartides which are alremzly being studied eoezimentally. Aad it may skedsome lilt on, or at lemst #ve some pre

' ' indication to, the solutionof the quœtion of what a sharp corne monns in small particles-

The atoec llmit in the present cont- does not Gtend as fa.r aq anattempt to start with a model which may reproduce the magnetization asdepicted in Fk. 3-1- Nobody hu ever approacxe this part of the problem.Also, it Ls not practical to study hundreds of lattice sites by the methodwhich is d-ribed hexe, and only nine spins are used, in what is supposedto be an iadication of what happens with the others. Allowing theamalfuctuadons in s'uck a small tpvticle' would have made them too strong forthis case, thus distorthg the physical picture for somewhat large.r padicles.Therefore, thermal agitation is not used, nor is thereuy attempt to changethe classîcal spins into quantum-menharical onœ, which should be done inthe atonzic 1''m5t. The spins are left to be the nln.qs'ical vectors used inmicromavetiœ, because the mn-m purpose is to study the limit of thex-nvnption of a physir-qlly =nll sphere.

The model Ls thus made out çtî poiat spias which are locnllzed i'a thelattice pohts of a bcc unit cell, whose cube edge is a. Only nine such spinsare considered, whck complete one unit celk The frst one of these spins islocated at the body centrez which is taken as the point (0,0,0) in Cartuiancœrdinates. The other ekht are at 1zcm, whea'e pç = (+1,+1, +1) is theposition vector for the ïth spin. The numbezing of the spias is Rcording tothe scheme shown in Fig. 10.2.

The exchange interaction is x-nmed to be non-zero only between nea-rest ùeighboursj wkich mpxnK between Sz and the other eight sphs. Theexchangeeaerr is taken as the exprœsion in eqn (2.2.25)., but J is replaced

234 ANALYTIC XCROMAGLIETICS

FIG. 10.2. The numbering sckeme of nine spins arranged ia a bcc lattice.

by the more flrnlliar exchange constant, C, accozding to the defnition ineqn (7.1.6), leading to

9C:2c.. = - - mu . zzuj?a1ï=2

(10.5.42)

per tmit volume, where zzu denotes the unit vector in the direction of thei-th spin. A similar expression was also used ia a calculation (332) wkichignored the maaetostatic energy term, but did have an interaction withan applied feld. Tkat case (332) assumed a surfve anisotropy, withoutany anîsbtropy in the body. Here only a volume cubic anisotropy is used,without Xa, which leads to the anisotropy enera per unit volume,

92 2 2 2 ; aEu = A-z rn.: m6 ''Hrzsrz:. + mamj. .*

i=1 ,

(10.5.43)

H practice (516) it turned out thaf this equatîon, as written, was usefulonly for positive Kï, namely for the easy axes along (1001. For Kk < 0,which means easy axes along g1:11), the accuracy of computation from thisrelation was rather poor. A much better accuracy could be obtained byrotating the axes to z', y' and F, with z' along g111) of the oziginal axes z,

&, and z, and viting eqn (10.5.43) in these coordinate.The magnetostatic eneror îs taken as the interaction of dipoles at the

235

lattice sites with the dipolar feld of eqn (7.3.21), before the introduction ofthe physically small sphere. This enerr is written in terms of the vectors

Pï j = Pï - PJ1 (10.5.44)Fhere py = (+1, +1, +1) is the position vector for the f-th spin, alreadymentioned above. With this notation, the maRetostatic energy, per unitvolume, in the partiçular lattice assumed here Ls

Tc REMANENT STATE

a 982$.J2, m: . mj (mf ' pf ,j) (my ' p;,,.) ssu = a - 3 s + :81 .2+y lpi,j.l lpi-j'lï=2 #=

(10.5-45)w'' here the term for ï = 1 is written separately, and is

(10.5.46)

The total enera of this system was minirlézed numerically for the 18' directions of the nine spins for C = 1.73 x 10-6 erg/cmj and for eithe,rM = l7ooemu/cms aad Kï = 4.7 x 1O5 erg/cm3 or Ms V 484emu/cm3S 1énd Kï = -4.5 x 104 erg/cm3. The frst of these cases has ihe physical

cspamrneters of iron, aad the second one has those of nickel, used in spkte.of the fazt that real nicke,l Eas axl fcc and not bcc structure. For eitker of

. these cases, the exchange enera is many orders of magnitude larger than''the other enera terms, if a realistic value is USH for the lattice constant,c.. Such a big dxerence makes it very docult to obtain any reasonable'accuracy in calculating the total energy, and aay attempt to rnl'nlrmlze thelènergy encouaters a large noise. The trick used (5161 for the mlimization' was to stazt with al unphysicvy large value of the cube edge a, for which' the exchange enera was only four or fve orders of magnitude larger thanïhe other enera terms. For such a case, the energy minlmlzation could bek

carried out with a su/cient accuracy. The value of a was then reduced,àztd the enera was miaimized again, usiag as a start the values of the18 angles obtained for the previous c. This procedure was repeatedp and6y elimiuatiug the necessity to compute the enera for angl% which wererather far from the minimum, it allowed a,a extrapolation of the minimumenergy state to the physically signlcant lattice constant c, of several â.

, . The result for positive Jf.1 was (516) that the lowest-enerr state was notone of uniform maaetization. It was a state in which mz. = 1, but mu forï''> 1 was not exadly 1, although they were all vez'y nearly 1. The actuallowest-energy confguration could be expressed by one angle, because to a*zy high accuracyit had zns. = ms. = msm = m8. = mzv = m4v = mzv =

148 and ma. = m4. = mr. = mg. = mcy = m5u = m%u = rzgv, withM

236

rzzz = .-maz of approvsrnately 10-*. For negative X'z, txe same relationswere cbteed in the transformed cocrdinxte system z?, #r, a'nd F with z'

along the easy nxn's to the nn-lqotropy. The angles were an order of magnitudesmaller than for the Xet > 0 cxse, which is to be expec%d because of thesame fattor in the assumed values foz M2. H Ieals to a dference of two!orders of magnitude in the average deviatzon 1om saturation, 1 - (mz).

ït has thus been proved that there is no cube whicà is small enoughfor the nniformly maaetized state to be the one with tke lowest energy,in spite of the fact that over shoz't ranges the Gxch=ge enera is xveralorders of magnitude large,r thnn the otker enermr terms. ln other wordqthe Wndamental theorem' of Brown, which haz b-m rsgorously provedfor a spîere and for a prolate spheroid, d- not hold for a cabe, aad willobviouslr not hold for an erension of a cube to a prksm. The mathemadcaldtfeence between a sphere and a cube is that tke demagnetizing Geld of anniformly magnetized sphere is homogeneous, wlkile that of a cube is not.ln a uniformly magnethed cube, theze is always a tzansverse demagnetidngGeld, which may be vez'y small compared with the exch=ge feld, but it isnever zero- Therefoa at least when the angle change continuously and donot have certain discrete values, there is always a s 'hght, bat fnite, tilt outof the uniformly magnetized state. The phyhcal sigecaace of thcs eefl'et isnot vez'y clear, aad is qutte controversial, depending on the view of whethera cube or a sphere ks a better approtmation for the behaviour of a smallmagnetic paticle in real lsfe.

Of eonme,, it ks not the matter of the extremely small deviation fromsaturation, whièh is negli#blefor dmcst a11 applications. The controversy isabout the paradox of Brown, described in seon 9.5. The same reasoningabout a non-uniform demagnething Edd applies also to a saturation bythe application of a magnetic Eeld, and leads to the conclusion that acube cxn never be stridly Mturated by any ânite, '//ZWJO= feld. And ifthe crystal does not start 1om mturation, the whole argument of sedion9.5 may not be valid. This possibility for the zuolution of the parHoxbad beem suggested (515) long kfore tkis calculation of the cube, and hadalready been discusse in the old rcviciw (œ2j of the paradox. It has beenstrengthened by many observations :517, 518, 519) of domes at the tip ofwhlRknzs, whch are not driven away by the applied feld.

These two problmmq aa not quite the same, and there is a diferencebetween a slightly incomplete sataratioa in the vichity of the corners, aada whole reversed domain there. These reve.rsed domldnn a2e nndoubtedlyreal, but they oaly apply to lave crystals, and theîr edstence may onlymMn. that some smples must be put in a larger feld before measuring thenucleation, as dkscussed in section 9.5. The slight deviations at the edgesmay or may not be suëdent to st.art the nudeation process lom, and thecube Xculation does not really cbange this contrcversial issue. It is stillthe same o1d problem of which is a better approzmadon for wwtl particles,

ANALYTJC MICROMAGNETICS

TlIB REMANRNT STATE 237

aad it should be remembere tltat micromagnetics results do agree withcerlma-n e'xllerimemts on mcll partides: including particlc with very oddsupes. Tkeoretically, t%e nudeation problem was also solved for kn4m'telylong prisms, which have a square (5201 or a recvngular (291, 521) cross-section. However, an l-n6nit.y is always suspicious in these problems, and ataxty rate, the inlnity in this case only evades, not solves, the problem of thesaturation near the Hges. A more recent numerical approRh is discussed insection 11.3.5. It hms demonstzated that tNe eAI:t of $he edges on nnmezicalresults is negvble, provided the discretizaûon is suëciently Ene.

A moxcation of the cube calculation (522) was 'USM to address theproblem of wheier the minMal enera non-llniform state ks due to thehigh symmetu of the cabe. !(n order to break this symmetry, spin number9 of Fig. 10.2 was moved fzom the position p: = -(1, 1, 1) to -A(:, 1, 1) ,with l having either the value àa or the value 1.25. For b0th cmse.s tke restlltfor lazge a w&e quite diferent fzom tàoK f or the previous em-, but b0thof them extramlated to the same results as before for realistic valuœ ofseveral i for a.

11

NUMERICAL MICROMAGNETICS

1.zt all the numezix camputations in mitzomaaetics, neazly all the com-puter time is spent on computing the magnetostatic energy term for thedl'Ferent maaedemiînn coMgurations which a2e being tried. It must be 'emphuized that this feature is independot of the computational method,and is a rstqnlt ofthe inevitable fact that the mmetostaticenera is defnedby a six-fold integral, as explained in section 6.3, wher- all other energyterrnq involve only a three-fold inteeon. For tàis renmnn it is importantto choose an eEdent and Rfrective method for computing the magnetostaticterm, while aay numerical Anxlysis method will do for the other tarms. Thedeseption will, therdore, start from this term.

11.*1 Maaetostatic Ene-rrMany of the numerical computations aze based on the method developedby Lcontej for computing irst a onedimensionat domain wall (301), andthen a twodimensional one g310).

In two dimpnsions, the wall structure is assumM to be independent ofthe dimension z of Fig. 8.1. The wall re#on, I=I S a and 1,1 S è, of thez&-plane is divided into Nz x Nv square prims, of <de

25 2al = - = - .Nv N. (11.1.1)

The latter relation lMits the pazameters to thf- which rxekkisf.r bNz = lNv,but tikis limitation does not Gect the geaerality of the method, becauœthe width 2/, rxn 1x extended arbitrarily into the domx.inK. ne badc ms-sumption is that witkin e-xrtb of the prisms, -c,+ IL K z S -c + (f + 1)A,-b'+ JL S tr S -: + (J + 1)A, the maaetizatioa does not nnn

For a constant mxgnetization ia each prism, the srst term in eqn (6-3-48)wan-qhes, while n .M ia the second term is a constant, widch cnn be movedin front of the integral- The htegrand then contains a.n algebraic Rnction,whœe inteval is HI)= ia prhciple-lt is thus possible to obtaia an anazyticexprœsion for the contribution of tuztb pzism to the magnetostatic potential,and the wholepotential is the summation of these coatributions over all the -prisms. lt should be noted tàat this l-%nlque transforms a volumochargecontribution, if it eists, to that of surhce charge on the four sudacesa the prisms. If the magnetization does not change between one prism

MAGNETOSTATIC ENERGY 239

pad îts neighbour, the contribution of a positive charge on one side ofthe sudace between them will be exactly cancelled by the contributîon ofthe negative charge on the other side of the same surface- But if there issome change, the diference between these surhce charges expresses, to a''f rst order in small quantities, the contribution of the spatial derivative; 6f the magnetization, which eventually converges to the contribution of:the ârst integral in eqn (6.3.48). Once the total potential Ls known, it caa.%. e substituted in eqn (6.1.2), and then in eqn (6.1.7), to înd the totalhagnetostatic enera. The latter integration can be exprused again asahotàer summation of the integration ove,r eacà of the prisms, which can.àzso be carried out aaalytically, once M Ls moved in lont of the integralsign. The result is (3104 that the magnetostatic energy per unit wall lengthin the z-d/ection Ls

m Nu m Nv. 2 a 2 l ,FM = L E J7 x >z(J, J) + MvLI, J)1 + .j. 5-2 )7 -4m(J - J , J:

' J=t J=l J'=1 J?=l

- J?) (J&(.J, JIMwLT, J') - MuLI, J)MuLI', J% + Cm (.J - T, J

- J/) l.&Jz (J, J)Mv (.r , J/) + Mu (./, JjMA (J:, J/)) ,

where Am and Cmare euluated from the above-mentioned integrals- Theqxpressions for these coelcients have been evaluated, and are given, in(3. 10). Some algebraic transformations have been implemented in order tolake these coeEdents more suitable for accurate numerical computations.1TJ hey are listed in tbis improved fnshion in (5234. It should be particularly, émphasized that the accuracy of these coeEcients is very important for a'ereliable computation. Experience shows that even a rather small inaccuracyin the coeldents caa,n lead the whole computation astray, and end up inCa completely wrong confguration. Another way to increase the accuracyi,s .to combine together the contributions from two neighbouring prisms,thus avoiding the subtraction of nearly equal numbers. The details of tidsmodifcation are described in (5244.

The four-times sumnaation in eqn (11.1.2) (which would become sixtimes izk three dimensions) is a manifestation of the long-range nature ofmagnetostatic interaction. Every change of the magnetization in any oneprism Wects the energy evaluation for all the other prisms. .A.s hms alreadybeen mentioned, khi!k property makes the computation Ume much largerthan it is for the other energy terms, but there is no way to avoid it.There were certain att/m pts, reviewed in (27(1, to approvlmate this long-

. range interaction by a local îeld. They were not successful, and only led to'.intolerable znistaku. It is Just impossible to substitute a short-range force

(11.1.2)

240

for a long-range one, ucept for a spedal case which will be disolqqfvl i:aSection 11-3.4.

The mos't importaat advaatage of this method, namely of writing tîemagnetœtatic enerr in the form of eqn (11.1.2), is that the coecients.A,a and Cm need not be eYuated over and over again with every itcationof the msn'lmlzation process- They are computed, and stored, only onceabefore the aztual computation starts. Any other method involves at leastsome computaticn equieent to these coeEcients, which has to be nztnqedout again for evea'y iteration. Thls repetion mxnr times ov& do% makea big dxerence. Therefore, all other methods which have ever b- usedjald whic.h ve reviewed in (2881, either use an împossibly long computationtime or go into rougk approfmations, or b0th.

Computhg X,x aad Cm only once also means that only negligible &m-pu*r <me is swnt on them. Therefore, it does not make any dîFerence ifthey are easy or d-llcult to compute, nor is there any reason to tzy to savesome tîme on their computation by introdudng tx'tain appremations, orother haccuracies. ln spite'of that, there lzave been mitny compuvtions,also reviewed in (9M), ia whicx the iategration ever the prismfacu was justreplaced by the âeld of a dipole at its centre. TMK presumed simpMcadonleads to the same restllt of eqn (11.1.2) with sliltly diferent value of .4mand Cm. lt is not a big numerical diference in the coeedents, but it canrnnve (288) a big difereace in the reults. And it is completely unnecesarywhen the expressions for w4,m and Cm are kaown and published, and it isno trouble at a11 to store them oace and for all on the computer. MorYver,(

many antkors Arm so convinœd that it does not make a diference that theydo not eve,n mendon if they use the correct coecieni of TmRoatea or onlyan approzmation for them. It is thus dilcult to evaluate properly manyof the reults in the literature, or to compare them wit,h enn% other.

The s=e vgument rnn J2mn be applied to another approzmation, usedin many two- and three-dimensional compqtaïons. Ia this appremation,the iategration over the muare (or tke cube) is replaced by the.feld at itscemtre, due to the charge on the snGltces atound it. Again, this methodleads to the same relation as ix eqn (11.1.2), only with somewhat dferentvalues of Arz and Cm , and agah it is quite mmecesar-r. TMs approzmar-tion has 13ee.11 justifed (299) by rqniming tbat it is easy to ckange it Foma cube to other geomeGes- But to meh at leastt it seems very strange toiatroduce an inaccuracy into a'calculation only because it is easy to. intro-duce the >me inaccuracy into aaother calculation. lt is true, of course, thatthe mGhod of TaaBonte is retricted to one particala'r geometry, and whenanother geometzy is needed, it is necxury to work out those integrals 1omthe beginning. lt may also be. necessary to ur.- certaia approfmations forgeometries for which these integrals Hve not beea evaluated. But there îsno renqnn at all to iatroduce haccuracies into cazes for which the accuratecoeEcients are alteady known.

NUMERICAL MICROMAGNETICS

MAGNETOSTATIC ENVRGY 241

The generxh-mation of this method to three dimensions, with prismsreplaced by cubes: enables computation in three-dimensional bodiœ, suckas prïsms.Tîe coeldemts for writing the magnetostatic Hteractioxs xmongcubes) and the magnetostatic enerr term, are listed 5n (525). A subdivision1nt0 cubes was n.ltkn urval for the study of fnite drcular cylinders (526),discnq.-d in section 11.3.6. When the surface detai.ls are not importantzor when the cubes are small enough, suc.h a subdivHon can be usM iaprindple for other shapes as well. Another genernlîzation of this methodyields (52% the coeëcients for a periodic domzu-n wall j but only for an.essentially Go-dimensional case in wlzic,h theze ïs no dependence on y ofFig. 8.1. It n.lpn indicated (52% a hrge saving in computing the four-foldsummationsr which nxn be used in the cmse of eqn (11.1.2) as well.

For a sphere: the A111 three-dimensionxl Tta-e.p has not b*n worke out,but two simpliîed cases ezst, for 'two difereat physical assumptions. Oneis a one-dimensional case, in whiG the sphere is sliced (2524 into planesalong the direction of one of the Caztesiam coordinates. The other problemis twodimensional, having a cylindrical symmetu <th no dependence on#. Eere the sphere of radius R is subdividu into Np x N. acdï-fomïdo,

I - 1 8 I J - 1 r JS - S (k-' aad f (à S p-, (11.1.3)Na 'r o Np r

where r and 0 are the polar coordinatœ, and 1 S I K N. aud 1 K J <Nr are integen. The magnetizadon is assumed to be constat h p-qztb ofthese quci-toroids, but for the maaetostatic enezgs M4/ sin 8(f , O andMe(.J, J) are txlrnn ms const=t,s for the integrationj whic,h then proceedsas for the prisms of ToRonte. The rœùlt is (253j

3 Nn J-1 #. N.caca , , g ,)m.(.r,,.z,)su = v X E 57 X'! ALI, J,1 , J )zn.( ,''S'r J-2 Jzwz z=z z'w.z

+ BLI, J, I', J')znr(J, Jtmo (J', J') + C(I, J, I ', J?)m:(J, J)rzr(J', J')N. Ne Ne

+ otz, J,z?, J'jmoq, Jlmdtzz,,zzlj + y'l )-! y! gstz, ..rz')m,.(z, oJ=l J=1O=1

x zrwg', J) + FLI, J, JJ)rw(f, J)m.(r, J) + GX, J, J')rp,p(.r, J)N. 1 N. (zz - 1)x mol1., J)j + 2 sin (J,) E,., (.r2 - J + à) ,E., sia (W (go -)

x' (m.(z, J))2 . (11.1.4)

242

The coefdents, obtained 9om integrations over the surfaces of the quasi-toroids, are listed ill (2531, with the typos corrected in (527J. They areexpressed as slowly-convergent sez'ia in Legendre polynorn-lnls, but then theconvergence is not important for coecients which are computed only once,before startiag the time-consuming minimizations-Taldng Moj sin 0 insteadof Mo as a constant in eack quasi-toroid maàes it possible to catr.'/ out allthe Gtegrations analyticxlly, but of course it htroduces a certain error byneglecting the variation of sin 0 over the range of such a toroid. H principle,this error is negligible if the subdivision is suëdently fne. In practice ityields quite a Mgh accuracy even for moderate No, as has been checkedby using this method for computing the magnetostatic energy of several

'

conkurations for wkich the result caa n.1m be evaluated Mnlytically.There is no other three-dimensional body for which these coecients

kave been calculated, mostly because the expressions become long andmzmbersome. A beginzdng of a calculation for a fnite circular cylinder,which has never been carried as far as yielding a practical fozm of thecoeGdents, is found in (259). Of course, it is always posdble to computea11 these coeEdents numerically for any given body skape, store them andthen use them f or minimizing the total enerae. Actually, such a numericalcomputation hms been carried out for a cube (263), clnl'rnsn g that it is simplerto do it this way, talthough complicated ralytic forms for the interactionenergy eztn be obtained'. For a dferent geometry, this method has onlybeen used in one case, Jscussed at the end of tMs section. Most of thosewho carzy out such computations still prefer rougher appro-xn'mations, orcompletely diferent numerical analysis methods'.

As has already been mentioned, other methods are impractically timeconsuming. This point must be emphasized again, because computationalpacHges are available nowadays that can be used to compute magneticîelds without even knowing what they contain. Also, there are specialconferences on magnetic feld computations, and for several years now theproceedings of eack of them has been a thicker book than this one is. Mostof these programs, aad studies of improved methods, rxn only be used for111ne,M' magnetics: whick means that they work only for cascs in whicheqn (1.1.3) is valid, but there are also many studies in which the magnetjcfeld is computed for ferromagneiic materials. Some of them obtain thefeld H from eqn (6.1.2), with the potential obtained either by a numericalsolution of the diferential equations with their boundary conditions of eqns(6.1.4)-46.1.6), or by a zmmerkal Otegration of eqn (6.3.48). Others obtainB fzom the vector potenual, briefy mentioned in section 6.1, which %M notbeen used in this book. Eowever, a2 these methods are devised 2528) forcomputing the îeld of a given magnetizahon confguration oaly oncm Most-of them can also be extended, without dieculty, to the computation of thema>etostatic energy of a Tven structure, but only if it Ls done once. 1n atypical energy minimization, this energy must be computed tkousands of

NMRICAL YCROMAGNETICS

BNBRGY MWJMIZATION 243

times, fzom a magnetization distribution which keeps changing with everyiteration. Present-day computers are too slow for this task) except for thosekho are ready to spend (529: Sseveral to many' CPU weeks on solving such aproblem. Improving the technique (530) 531j did not change this time-scalebuëciently to make it more usefal.?. In one recent case (532j the numedcal analysks technique has returaedto the idea of LaBonte, of computing (and storing) a1l the coeëcients forthe magnetostatic energy term before starting the actual iterative processof the enera Dinimization. It has not been presented as such, althoughthe authors should have Hown about this idea, and the computationsalready carried out by using it in mauy problems. 1n this case) the b%icdiscretization element is a tetrahedron) for which the LaBonte coeEdentspre computed numerically (532) by two dxerent methods. It is clnsmed tobe a veur general method, but it has only been used for a certain prism,,for which cubic elements, with the alreadyzknown coe/cients, could havebeen used just as well

1 .2 Enerr M'nlmlzat on

Calculation of the other enera terms is straightforward. For the anisotropyvnera term, integration of densities such as eqn (5.1.5) or (5.1.8) is justbroken into a sum of integrals over individual prisms (or cutes). And since1he magnetization is assumed to be a constant in each of these su'bdivisions,each integration is equal to the area of the prism, or the volume of the cube-.'I'he same applies to the enera of interaction with an applied feld) if used.

The exchange energy can be obtaaed directly 9om eqn (6.2.45)) afterbubtracting the energy of the uniform state, as is done in eqn (6.2.46).T .here is no contribution 1om the body of the subdivisions, where theneighbours are parallel to each other. Therefore, the total exchange energyi's the sum over a11 the surfxces between neighbouring subdivisions) of aneoression similar to eqn (10.5.42): which is derived directly from the theorykb. f Chapter 2. LaBonte (310) did not do it this way. lnstead, he started fromthe classical expreasion of eqn (7.1.4), and approvimated it for small angle,practicany working out the derivation in section 7.1 backwazds. The result,however, is the same. For curved subdivisions, such as the quasi-toroidsùsed (253) for a spherej the procedure is essentially the same. It is alsothe spme for a one-dimensional study (53% 534) of an infnite cylinder, inwhich the magnetization depends only on p. It has been presented as a'new technique, and named the latomic layer model', in order to (Iistinguishij 1om micromagnetiœ, that these authors understand to apply only toanalytic calculations. The latter case also Ieft out the magnetostatic energy''tèrm altogether, becaux they deal only with the curlhg lmodel'.

, ' The foregoing should be a su/cient outline for writing the total enera

t in a form which ca.n be coded as a computer subroutine. There are veryeEcient computer programs for minimizing an expreasion in a subroutine

244

which depends on several parxmeters, and it may seem at frst sight thattkere should be nb diiculty in Iainimizing the total Oergp Eowever, theseprograms are limited to ml-m-rnization whh zespect to a rehtively smallnumber of parameters, usually of the order of 10. They Onnot be used form'pnl'roizizg the enerr with respect to the maretization vector in p-qzth ofthe discrete subdivisions in the present context, because thdr numbe,r istypkally in the range of thousands or tens of thousuds, aud even more-There are two methods in the literatuze for the minimization iœlf, au.din both of them the expresion for the enera is frst used to compute theeFective magnetic Eeld, desned h eqn (8-3.41), at eath of the subdivisions.This feld, Hem, is essentiuy the derimdve of the eaerr with respect tothe local maaetizauon vector, and rxn be evaluated numerically for eachof the subdivisions directly 1om the energy exprasâion, without going backto the do6nstion in eqn (8.3.41).

ln one method, u>d in e-g. (253, 310, 3231 326, 327, 337, 524) and others,the maretization vector in 1nn% subdivision is zotated to the dizection oftkis feld, mn', atthat position-After swepingthrough all the subdivisions,the mxr-mum angle of this rotation i.a any one of them is compared with apreset tolerance. The procv of rotating the set of m(J, J) poht by pointthroughout the grid is continued uatil tMs mnvl-rnntm augle is smaller tlmnthe required toleranœ, at which stage eqn (8.3.40) is obviously fulved towithin this tolpmnce. It can be showa that in this method the energy alwaysdecreases from one iteration to the nert. This property is an advaatage forrelatively sim' p1e energy manifolds, and at least it ca= never go wrong ifthere i. s only one miMmum. lt may be a disadcantage if there are at leasttwo energy minima, with a certain bazrie.r between them, in which cmse astaz't in the vicinity of the higher minimnvn may converge therey withoutevez crossing over to the lower minimum.

The other method: used 1, for exxmple, (299, 324) and in many ofthe numerical computations which will lx d-lpztussed laterv is to solve nu-made>lly the dynamic equation (8.5.50) or oae of its vadations discussedin section 8.5. For static problems, such ms the structure of a stationaxydomldn wall, or the remanent state of a pazticle, a dxmping parxmeterfor induhon in that equation is ehosen arbitrarily. The main advantkage ofthis method is that it ks readily adapted (32% for real dynxmic problems,such as a moeg waz, or the variation of the magnetization after a mxg-netic feld ks applied. For stddly static magnetization coafgurations, thismethod does not seem to have any admntage over the previous one, orat least none %J'x been claimed in any of the publications desczibing it. ltmay n.ltkn be time xnsuming, becace intermediate stagœ evolved in timeare of no particular inteest in this b''nd of computation. In an improved .

variation of this method (532) , the dynnamle equadon is m'itten for eve.ryone of tbe subdividons separately,' aad not for the whole sample, and thetime-step is adlusted in ezach iteration to be ms large ae possible) as long as

NUMERICAL MICROMAGNETICS

ENERGY MTNIMIZATION 245

the energy is not allowed to increcac in any single step. Since this purposeis automatically aeieved by the other method, descdbed in the foregoing,the advanvge of this method is doubtful at best: for any statîc problem.Of course, it is irreplaceable when the real dynamics is sought, except forthe case of a wa2l moving uriformly at a constant speed, for which thestatic techniques can be applied, aas discussed izt section 10-4, and as donein, for fwvitm ple, (3371 . The best description of the details of integratin.g thetimodependent equation cxn be found in (5351, and a comparison of thediferent methods for this integration is given in (536) . .

ln either casej the boundary conditions have to be enforced by choosingspecial rules for the magnetization in the subdivisions on the surface. Forexampleo when a domain watl is supposed to end in a domain on both sides,the magnetization in the frst and the last row of prisms is always kept S'XE':Iin the appropriate directions .g310q. The boundazy condition t'?M/t'?n = Ocan be enforced by adding extra subdivisions just outside the materiall inwhich the magnetization is (5374 a mirror image of those just inside, or byother (310) methods.

The convergence of either method is quite slow. lt has been noted (538)that a much hster convergence can be achieved vez.y ofken by groupingsubdivisions to be changed together at each iteration, instead of settingthe magnetization in one subdivision at a time. Therefore, the number ofiterations, and hence the computation time, cnn be much reduced, if acertain pattera can be îdentîfled, for a cooperative magnetic change, or amode, in a large group. This method, however, has only been used in aspecial case (538) of a domain wall motion, for which it is rather eaEy toidentify the parts of the wall, and make them move together. ldentWingthe relevant Cmodes' in other caEes is not that simple, and it still takes adeeper study before this method may be more generally used.

A more dratic approach of this knd is to group together several of thesubdivisions pevmanentlv and treat them as one entity: by aEsuming thatthe magnetization is always the same în each membcr of the group. lf donefor the whole sample, this assumption only means a crude mesh, which iseasier to solve than a fne mesh, but leads to a lower accuracy. However,when this technique is used selectively, it may save computations withoutlosing the accuracy. lt was actually used (323) izt the computation of thedomain wall in very thic,k flms, for which it can be safely assumed thatthe magnetization varies much more rapidly near the surfaces than in themiddle of the flm. Therefore, a fzte square mesh was taken (3234 only nearthe surhees. Farther away fzom the surfaces) several squares were groupedtogether into rather elongated rectangles. Of course, thereis a big dlFerencebetween the qualitative statement that the vaziation is more rapid'near thesudace, and the quandtadve choice of a particular size for the rectanglesaway from the surhce. This particular case was justifed by its passingthe self-consistency test of section 8-4, which makes it at least better than

246

some # ld guesse, which are also being published. But this test nltn' oaly beapplied after a1l the computations are done. There should be some wayfor .a quantitative justlcation of the use of this technique, preferably before ,.

startiag the main part of the iterations, but this part has never been done. '

This drawback is only one of several unknown and unestablished points .aud assumptîons which are just being used in micromagnetics computationswith no justifcation. ln the old days, when computer resources were limitedand expensive, programmt-ng used to be approached much more carellllythan in more recent years. lt wms taken for granted) for exn.rnple: that aprogram must be checked by mmniag at lemst one case which cltn be solvedMalytically, and compadng the results. Of course, it is not practical aaymore to do it for every problem, but some sort of checkng is essential, even i

if the full three-dimensional problem is studied without approzmations,but even more so when some Ssimplifylg assumptions' are introduced. Theother extreme of having no checks at all is too dangerous, and its restllts arenever reliable. The computer is a very useful and mwerful toolt and it cAndo'wonders if properly used. But it is certainly not asubstitutefor thinking:and it is too common amistaketo let the computermake all the decisions. ItbnA become much too eas'y nowadays to write a program and run it, so thatone often wonders if certain published results refect some physical realitsor are merely the eFect of an overlooked error in the programming, or inthe logic leadiag to that program. It may also be just due to an approxymation which the progrn.rnmer does not stop to think about, or which mayhave been copied fzom another work, in which that approzmation is justi-fed, and yet may not be jastifed in the new context of another particpllxrproblem. Several such dllculties ic the paœticular cmse of micromagaeticscomputations, which stiz await a solution, are listed here. Some of themmay have been addressed, or even solved, in some of the studies. But theyare not mentioned in the publicatioas, which may be because many of theresults ate being published in conference proceediags with strictly enforcedsize limits, in which the most interesting pal't is often omitted.

NUMERICAL MICROMAGNETICS

1. The size of subdivisions is chosen arbitrarily in most of the reportedcomputations, and in many cases this size is not even mentioned, asif it mere an unimportant parn.rneter. No clear-cut criterion is known,but there is an obvious guideline between two lsmits. On the onehand, the mesh should not be too fne, so that the approzmation of acontinuous materialpdiscussed in Chapter 7, is still valid. On the otherhand, the mesh should be suEciently fne to allow the magnetostaticeneraterm to developfully the complicated structures that it usuatlypzefers. If the subdieions are too crude, the magnetostatic energyis too high, instead of becoming negligible as it normally isr besidesthe possible introduction (53% 5401 of discontinuities, and convergingto (541) a wrong result. A cntde subdivision may be adequate for

ENERGY MINMZATION 247

ca:es such as (542, 5V), when the walks aad the domains are studiMwithout the detvs of the walls. But it Ls ceztpimly not justled whenthe wall detvs are neede, such as in (324) where a 10m thick'l= is divided into 128 prisms, making the prism size about '?8nrn.Tlzis prism size Ls cn order OJ magniinde larger tha.n in some of thepredous works, and it is only used in order to elxim some rœults forthicker Slms than ever studied Gfore. This article (3242 dou not evencomment on tMs cxoice of the prlm ssze, and it ha.s other fault.s too,such as. not even mentioning whether the mngnetostatic enerr termis computed as in. sedion 11.1, or by some approzmation.

The safest method Ls to compute for a certnin mesh, then sub.divide it further and repeat the computations, to sœ the efect ofthe fne.r mœh. This proMure is almost standard in less complicatedcomputations. Thus for vample in (530J the accure wms càeckefor dferent meshes by eompxhng tlle Eeld of a saturated sphere withthe known analytic rcult. This check is certnx'nly much bette.r thanno check at all, but it is inadequatej because the âeld of a Mturatedspheze may not be a good memsm.e for the feld of the more complexconfgurations. Fœ the actual enera mlnlmlem.tion in (530) , 2639 ele-ments were just Gosen, without kying any other subdivsion. h a Ititzmodel for a sphere, for wbich the magnetœtadc enera was solvedanalydcally, and oaly Ge excxange term was computed numerically,it wms possible (M0) to start with a 31x31 subdivision, then incemseit to 33x33, and fmazy use 66x66. A much preferable approach isa systemauc study of the Kect of subdivision Mze, ms done i.a two(544) and three (545) dimensions, but it is not always practical forproblems in whic,h the compute.r resocces aze pushed to their maf-mnrn lîmlt ms tqkon happens in maaetostatic computations. Thus, adetailed study of the Arzturacy has been reNrted for some numericalcomputations (532, 546) of the Loonte eneldents, but ne- of thecomplex msnîrns'zmtion itself. Of course it Ls important to mnlre surethat Ge cœëdent are accurate, but it is not suëdent- Even in ac%e such ms (532) whic,h is compared directly with experiment, it isalso necessary to check that the rest of the theory is done properly- ltmay be advantageous (54% to allornate Gtween a cox- and a fnemesh, but the mnin point is to end up with a suKdently 5ne meshz

2. The approvh to A'n6nlty is never well defned nttmerically. It r.n:n

only be expected to be taken suldeatly hr from the main body ofthe computations, but there is no clear-cut guidplsne on how far issuldently far- Agaia, it is necessazy to try more tlzan one wzue, andyet sueA a check hms not been reported in any publicadon. H domainwall œmputadons, for uampleo the domains are expected to start at2 = :i:a of Fig. 8.1, where a should be large enough to allow a G'11

NUMERICAL MICROMAGNETIOS

spread of the wall. And yet, this a is just Gosen arbitrarily, withoutaskng wkether it is large enougkl and witkout trying t/o see the elec'tof usinz a latge.r c. In a numerical rlution of tke rl-earential equadons(6.1.4)-(6.1.6), tphe behaviour at infnity is an Hportant boundarycondition, but shonld the snfnit.g be talcen as 5 times t'he radkus ofthe ferromaaetic body, or 50 times, or what? H studying a spheze,it Ls mentîoned that the potential ne not be computed only in thesphere, but also in a dmuch larger surrounding re#on of free spaœ;15292, but no indicationss #ven on how lazge is Cmuc,h llger'. For theadual computations, 2639 dements were used in the sphere, and 7534outside, and if aly other numbers were tried, they were not reported.

3. Self-consistency tests, such as those diMussed in seckon 8.4, are vearyMportant to c,he that the results make senœ, and are not merelythe result of some mistxkMy or of wrong appzozmations. Using themis the only known way to dxa'm that for a certain razge of the ilmtMckness, the neseded third dimemsion cannot chaage t'he computedtw>dimensional wall enerpr very œnsiderably. No improvement ofthe computational accuracy by itself rxn reveal this information, ève,ni.f those computations are 100% reliable and 1ee from errors. Thedx-lcult.y ks that these tmts have only bee.n developed for static, orfor nnt-forzlzly movhg, domain walls. It is, thereforev lmportant todevelop similr tests for other caso, and to ure the es-ting testswherever they applp Any publication in which the self-consistencytest is ignored should be suspcted and not relied upon. The testis ocntïtcffre, and an article sucà as (3021, which only >ys thatthe results were 'tested' this way, vithout specifying the ruultingnnTrnbersj looks straage at best.

4. The way computational results are presented is the most diëcultprobl= in trying to extrac't Mormation 1om published results. Thestm.ndard methM for preenthg twœdimensional walls ks by plotssuch as the one in Fig. 8.3. They do not reveal all the fme details,but at least they give a good idea of the main structure. For thr*dimensional structures, this method is desnitely not good enougkbut no better way has been developed yet. The three-dimenshonalpicturœ made out of twœdimensional arrows in (529, 531) are incom-prehensible. Even reladvely simple structures, such as those of (526),or of others which wlll be discussed in the next section, are not mucàclearer, althoush a cuà 1548) can sometimes help to see more details.Sucx plot are relatively easy to follow in the or= presentations incolzfeences, when diferent direcdons are shown in deerent colonvstbut these colours az'e usually lpst in the publkshed procrMvh-nr. Theproblem will be at least partially solved if thee fgurœ are pnblishedia cololm but there seem 'to be some technic/ deculties, whic.h also

ENERGY MINIMIZATION 249

apply to the presentation of certain experimental results, such as

(549j. Although there are already several publications in colour, e.g.(550, 551, 552), ihey are still quite rare, and the whole problem needsa more drastic solution. Actually, the best way to present numericalresults is to build an analytic approximation to them, as discussed insection 8.2. This method, however, is ver.y decult, and has hardlyever been used.

5. The convergence criterîon for termînating the iteratîons Ls the lemst-def ned parameter in micromagnetks computatîons. Dxerent authorsuse dxerent criteria, and their values seem to be chosen arbitrarily.There is never any attempt in the publication to just# the valueused in a particular study, and in mn.ny cmses this number is noteven mentioned there. At a frst glance it might seem that a ratherlarge value of thîs criterion is adequate if only a rough estimationis wanted, but experience shows that this criterion should be m'tzcàsmaller than the requbed relative accuracy of structure or enera. Inmany cases a choice of1 say, 10-4 may converge to a structure whichcan be changed drastically if the computation is continued with aconvergence criterion of 10-6. In one (unpublished) case, the energyof the system changed by about 3% when the maxlmum angle waschanged 1om 10-4 to 10-6. lt may not be fhe same change i.l othercases, but the mere fact that it cazz happen should warn workezs thatthey must be very carefal. More than one value should always betried, and it is wrong to rely on any one guess.

6. The eventual check of evezy theory in Physics is its agreement withexperiment. J.u the cmse of micromagneticsf however, such a check ksoften prematurej and may be more misleading than helpful. It shouldbe borne in mind that this theory, in its present form, is very muchoversimpl/ed, neglectiag importaat fadors such as magnetostrktion,surface roughness and crystxlline imperfections, discussed in section9.5. The efec't of surface anisotropy is not knowp but it is likely to af-fed strongly the experimcntal data. These Xects are ignored (553) inmost numerical computations, although they are easier to introducethere than into the analytic studies. Therefore, an agreement of theresults with some experiments is likely to be a mere coincidence (554)that only covers some computational errors, espedally since thesecomputations often make an arbitrary and unjustifed choice of other(553) parameters, such as the Jl.nl'sotropy and exchange constants. Thereal challenge of computations at this stage is to do them correctlyfor an ideal particle before considering real particles.

Because of a11 these dilcultiœ and uncertainties, many of the results.n thé literature are (5541 doubtfal. ln the next sectionl some results thatreem more rellable than others will be summarized, but even these results

250 NUMERICAL MICROMAGNWICS

may change with newer reseaz'ck. Detils which cannot be given here c>n .

l)e found in the cited pttblications. '

11.3 Computational Results11.3.1 Domain W'tzlùsMost of the computationz results for statiG 180* walks have alrpmz!y %enltsted i:a section 8.2, and will only lxa brie:y redewed kere- There is ùeithe,rGperimentat nor theretâcal information ou the wall structure oz energin bulk materixlK. We Yt Gimation of the wall energy in tke bulk u'

1 C f the Landau and Lifskiiz 'still based on the one-dimensional calrm nt on owall, dœcribed in sedion 7.% whiey is most probably wrong. Computerreourcw are e%austed at aa iron Gl= tkinbness of about 3 to 4 Jzm and .tkere Ls no way to tell what kappens for a large tidckness. At about tltnxttkirvnerxs there s= to l)e a kansition (3D) 1om a thin 6lm confgura-tion, where the curve on which Mz = 0 is shaped ll1ep the lette,r tC' to aHlFereat confguration for which that cmwe is shaped lîke the letter CS'. lftkis transition eists, it may be an hdication for the watl structure in thebulk. Howe-, even in the tMckest Glms which could be computed, therewas no Gangeover from Ge C- to tke S-type structure- The ener#es oftkese structura bdng very nearly the s=e at t%h thickness, either oneof them could be obtained, depending on the symmetry of tke startkgconîguration (323).

For aa intermediate tkckness, betw- that of a few Jzm axd that ofabottt 0.1Jzmj or somewhat less, the situation is qaite cleaz-There are xmetwo-ap'menskonal computations of the wall stmcture and ener, whic.à #vethe results ms described in section 8.2. There is n3= a thr ' enKional 't

computation (555) in this thicHus region, nxmely up to a tlzio.knv of E5OOnrn whëc.h is taken to be alzeady tbulk'. But tkis study usœ a rough? .apprormation for the maretostatic enetv, and very =de subdivisions.For still tkinner flms, a three-dimensional computation beomes essental,but none of tke published resul? is dear-cut, because they are based on.an approzmation for the maretostatic energy tm that ca,n (52% beavoided now. Ia particular: no computatkon skows a transîtion from theBlock wall to tke crosmtie wall at a thickness whkh may be compatiblewith tke experimental olerntions on metallic Elms. A changeover fromthe two-dimensional BloG wall to a symmetric Néel wall was sought (556)but not found. Instp-'td of a well-deâned transitionj the computatkons (5565enountered a wide thielrnes region in wikic.h noae of thcse two strttctureswas stable. This rpvnlt is not surprising in view of the fact that cross-tiewalls are observed in thni size re#on, bat this structure cannot appeamina computation whicà is constrxlned to be two-dimensional. lgnoring tMsex-pedmental fac't, some otker explanation was sought in (5561, with anattempt to enforce that tramsidon by an application of a magnetic feld.

COMPUTATIONAL RESULTS 251

In the case of ferrites, the periodicity along z of Fig. 8.1 is usuallyrgplaced by a ldnk in the wall, whic,h is 1% pronounced than the cross-tiejsructure of permvoy flms. For this case, many more details have beenéùmputed and compared with expeziments (5571, but only for a small parti' f the wall in the vicinity of the change of wall chirality along z. Some' ;1è. o-dimensional computations (558, 559) point the way on how to take:4t'dto account the dependence on the thizd dimension, z, by summing over'éhe periodicity along that direction. However, these computations did notyeàlly address the problem of a cross-tie wall, and were only done for the'# priodic structure Hown as the tstrong stripe domnsns'. The theory of theXéel wall in very thin flms is not very clear either. In particulaz, there is.Ep computation of such a wall whic.h obeys aay self-consistency test-.A .'

A wall must develop aa e-x-tra asymmetry when it moves, as explainedis section 10-4. This asymmetr.g can be clearly seen in numerical solutionsLdseqn (8.5.52) such as (56% 561). These computations have also been done$62) for the case of a two-dimensional wall moving through a region inb'' hich the anisotropy constant changes, as in ssm-lln.r, but one-dimensional,)7'c)51. culations discussed in section 10.2. These studies, as weE as the oneir'hick addruses the efed of eddy currentz on the wall motion (5634) are.;f.iestricted to one- or twœdimensional structures in rather thin flms. The'J . .

fnclusions (5634 about a thinkness of 1 #m; for exxmple, ire not based onbqmputations at this :1m thickness. They are actually obtalned by scalhg6hé. physical parameters of the material, and computing for 0.1 yzt flmi:ickness, without paying any attention to the changîng subdivision size,)7or convergence criterion, or any of the other points Iisted i:l the previousiection. A much higher accuracy for thicker fllms was repo ted (337) ) butf,hat study was only for uniformly moving walls, aud could not be extendedfztl large velocides. It could not be extended to vezy thick Alrnn either,because of the same limitations of the computer resources, mentioned initke foregoing f or the case of a static wall. A dynn,rnic study of a 2 yzm thîck,

; , L ) jj j jjje wauX'lm showed (5644 a transition fzom the C to the S s ape obtwjmcture, mentioned in the foregoing, but this result was obtained with a

;'

1. nde subdivision. Similarly, studies of the motion (565, 566) of a part of aiwhree-dimensional wall is also subject to the above-mentioned limitationsfor the statics of such a wall. A particularly hteresting trick (567) is to apply,*)p. eriodic feld to the computed wall to stabilize its pe iodic structure.llt'. . .ks has been mentioned in sectfon 8.2, a very good approfmation for thejh# all structure in intermediate flm thic-kness cmn be obtained by enforcingiero magnetostatic energy. Such computations are much fmster than anyij f the others described in this chapter, but they involve the problem ofG'tting T' . M = O together with the constraint 1Mi = Ma- Ia a mriation

' '

$f this method (5684, used for three-dimensional computations, the frstsïonstraint V - M = O was mainvined at each grid point, but the secondtme, 1M1 = Ms, wms not. Instead, iM1 was allowed to change fzom one

252

point to another, with an eventual convergence towards the same <ueeverywhere. This technique w> later ignored, and never used age.

Some computaïols of domin wall structuzes lkave nlnn been utendedfor looldng into the aualysis of magnetic force miezoscopy (MF'M) data. lnone extreme (569, 5701 571) the meuured magnetic confzgration was lzkkenas Onstaat, aad the resulting maaetic conâguration in the AGM tip wascomputed. ln the othe,r extzeme (572) the fne details of a two-zimensionalwall were computed, - into account the feld due to the measurlg tip,but the detalls of the tip magnetization were neglected. TMs magnethationwas assumed to be coastant within the Lspheriœk tip, and v?ms allowed onlyone degree of freHom, for it,s directlon to be zotated in such a way that thetotal enerr of tip aad sample is a minimum. Such aal approimation maybe justifed by m-urements of the magnetic feld in the vidmity of a sltamMFM tip, which seem to be well approx-imated (5734 by a (dipolar) feld ofa sphere. Of course, these measurements were made without a sample, andmar not be indicaove of whai happens when the sample is nœby. Mozedetailed computations of a.n enerr minimum of the total energy, allowinga mriable magnetic dlen-bution in both the sample and the tip, are hintedat h Ref- 19 of (572), but neither the detils of the computations, nor tkeirresulk, were ever pubishM.

The conclusion Fom (572) was that the m-mnurement hms a negli#bleeFect on the measured pattera. An opposite conclusion, that tàe estenceof the memuring tip has a very large efect on the measared magnethationpattern, wms re, VACSII in 15311 . Tke rexson for these diserent condudons isnot Hown.

Finazy, it should be emphasized that this discussion h% been limitedto a 180* wall, ignoring other wallsl such as 90O onas. Wken the latter arestraight lines, their computation is rather similar to those prescted here-There is also a large number of worH on the statics (330) aad dynnxnsœ(574) of euaed wll.1lq, which are outside the scope of this book.

11.3.2

NMRICAL OCROMAGNETICS

SphmThe theoretical remanent stateof a ferromagnetic sphere has been discussedin section 10.5.1. It was proved there that belcw a œztain Tcritical' Teusjthe lowest enerar is that of the uniformly magnetize state, but this radiuswms only given there ms a reliable lower bound. The aaalytically Ycuhtedupper bounds tnrned out to l)e much too large; an.d their evaltiation hadto be done by nnmerical methods.

Using the method dacribed in sKtion 11.1) of subdividing a sphere intothe qnMi-toroids def .ned by eqn (11.1.3), the lowu-energy confgurationwas computed for one case (2532 of a uninm-al xnîmotropy aad two cmses (509) .

of a cubic n.nisotropy. Tt should be emphmsized that becuse these computartions are constrxx-ned to cylindrir4lly smmetric confgrations, thc resultis in prindple an upper bound to the enerr of clJ possible confgurations.

COMPUTATIONAL RESULTS 253

Therefore, the critical radîus whkh they imply is also an upper bound toh dius ln the two computed caies for cubic materials the uppert e true ra . ,bounds thus obtained were only about 40% hrger than the lower bounds.Together they thus deine the critical radius to within the accuracy withwhich the values of the physical parameters are known. For unieal cobalt,however, the diference is much larger. The computed upper bound (253)of 34.1nm is smaller than a11 upper bounds computed Vfore, but it is stîllthree tîmes the value of 11.5nmj which eqn (10.5.34) yselds for the physicalpara.rneters of this material. It is still a rough estlmation, and it is not clearwhat happens i'n the case of very large anisotropies: for which the upperand lower bounds of Brown are particuhrly diferent from each other.

The magnetization confguration just above the critical radsus is madeessentially of two curved domains, althoagh the word may not be properin this context, because the $wa11' between these domalns extends over anappreciablepart of the sphere. These domains have a cylindrical symmetry,not only in the computations for which this symmetu is assumed, but alsoin a full, three-dimensional compuvtion (52% 531j wîth no constraints: atleast in as much as it is possible to see i.n. the published confguration.lt seems that the whole confguration is very well approfmated by theRîtz model (462), as defned in eqn (10.5.40). The latter was originallydesîgned (462j to study the magnetization reversal by the curlsng modebeyond the nucleation feld. But since the study of' curling had alreadybeen forgotten, this confguration was presented (529) lus a new typey calleda tvortex' structure. The above-mentioned two domains should be imagînedas oppositely magnetized, in directions parallel to an easy anisotropy ao-ds,and the wall between them is mostly magaetized in circles.

If the anisotropy mnishes, the inner domain is tmiformly magnetized, inthe direction z of the previously applied feld, or very nearly so. The outerdomain is mostly magnethed in circles, namely Mç is azmost equal to Mstwith a small tilt towards z. ln both domains, the cylindrical componentMp is very small. The average Mz in the remaneat state is quite close toMn for a radius Just above the critkal value, and is rather large even forconssderably larger radii. Al1 this description is actually based only on thosecomputatîons which assume a cylindlical symmetry (5754 to start withpbecause the results of the 111, three-dimensional computations (529, 531)are not very clear in the published fkures. The latter were tîme-consumingcomputations, which approached the sphere as a limit of a polyhedronwithvery many faces, for which it was diëcult to obtain a suëcient accuracy forparxrneters known 9om the analytic analysis, such as the nucleation feld.lt should be noted, though, that it is still an open question (576) whethera sphere is a better approfmaiion than a polyhedron for the real physicalsituation, when St comes to very smâll particles) which do not contain verymany atoms.

The direction z doG not have any meaning when both the feld a=d the

254

anisotropy are zero. Therefore, the desczibed two-domain structure has thesam: energy for any direction; and cnn be rotated from one direction toaaother without azky enerpr barrier. Therefore; a,n infnitesimal magnetkxfeld should be able to tura a large part of the magnetization into thatfeld's direction. 1.n other words, the initial susceptibility îs infnite, at lemsttheoretkally. Suc,h a.n infnity (or a very large value in a non-ideal cmse)could be very useful for transformers, or read heads, if it were possibleto make a suëciently good approvimation for such smallj and isotropic,spheres in real life. This idea is not a complete fazdasy, now that ve,t'ymtn.ll particles, with an almost spherical shape; have been mnrle (473, 474).Their =isotropy is not zero, and they may not be small enough yet, bitsuch further steps may still be possëble.

For such a,n isotropic sphere, if it is smaller than the critical size ofeqn (9.2.29), the nucleation should be by coherent rotation. The nudeation 'âeld for this mode is 0 according to eqn (9.2.10), because Kï = 0 and

'

Nz = Nz. In this case, the Stoner-Wohlfazth model of section 5.4 shouldapply, with all the arguments as presented there. The whole magnetizationcurve then. consists of three branches. First there Ls the Iine Mz = Ms,of a saturation in the ea-direction, fl'om a hish feld down to zero feld.At H = 0 there is one jump to a saturation in the -z-direction, whkhis followed by the branch Mz = -Ms for al1 negative felds. Tlzis curve isreversible; namely it is followed again for a feld increasing fzom negativevaluei The coercivity is thus zero; the remanot magnethation equazs Msland the initial susceptîbility is izdnite.

If the radius of the sphere is just above that cdtical size, the remanent. '

state consists of the two domains described in the foregoing, for whichMr = (Ma) < Ms. There is still no hysteresis, and the mn.gnetization cuzveis completely reversible, according to the collective information from Ritzmodels (462, 54% 57% and numerlcal computations g52% 531, 575). Thesaturation Mz = Ms is followed 1om the high feld, down to the curlklgnucleation feld of eqn (9.2.28)7 wbich is positi'lle in this case of Kï = 0.Then there is a cuzve, which is nearly but not quite a stzaight line, leadingfrom the nucleation Eeld to a positive remanence value, Mr. At H = 0 thereis one jump to -Mr, and the negative part of the Inagnetization curve issymmetric to its positive part. The coercivity is thus zero, and the initialsusceptibility is still infnite) although the initial jump brings the averagemagnetization only to Mr, and not all the way to Ms. With increasingradius, H,v incremses, a.nd Mr decremses, until at some size the line from thenucleation point lead.s to the point (Mz) = 0- For this sizej azkd larger ones;the infnity disappears, the initial susceptibility becomes 'Fnste: tending to4*/3, and Mr = 0. . .

The theory is much less developed for Kï # 0, and defnitely needs morestudies. There az'e very few computations (529) 53117 and even foz them mostof the details have not been published- They are a11 for a sphere whose

NUMEMCAL MCROMAGNBTICS

COIXUTATIONAL RESULTS 255

radius ks such that Mr1Ms gs 0-2. It seems that when a uniafal anisotropylsi added to this sphere, in such a my that the nucleation feld is stillJositive, Mr v=ishes. It i. s rather diëcult to understand the reason for thislkfect, and even more diëcult to End out i:f this behaviouz Ls typicaal, or îf itAmplies only to a special case. There is no hysteresis around H = 0 in thesetomputed curves, and the coerdvity is zero. There is, however, a certainkysteresis loop for numerically large, positive and negative, applied Eelds,pear the nucleation and the approack to saturation. A similar hysteresisfear saturation has been observed in single-crystal iron Jlr?.s (578) . lt wms'then presented as a vers6cation of a certain theory of phase transitions,yhich predicted that this hysteresis should only ezst for cubic anisotropy,ààd only for a feld applied in the (111) direction. The theoretical efstencebf this phenomenoq in unlnMal spheres defnitely proves that there mayube mechanisms other than this phase transition that ean account for its6. bsermtion, but the details have not been worked out.:. .

M these resultsj as well as /1 of chapter 9: are lirnited to the caseùf xzero sudace Hsotropy. The latter rnny play a,n important role in realjarticles, but itz theory has not been suëdently developed yet. Except foripme approfmations for trivial cases, the sphere hzp.q only been studied.,746) for a certain fo= of surface anisotropy. Foz this cmse, the nucleation6,e1d by the curling mode can be emluated analytically. lt ictually involves(06, 553) changing only the value of q2 in eqn (9.2.28). Aùother mode,r'èpladng the coherent rotation whkh is not an eigenmode once such a,llHisotropy ks introduced, rztlled for a numerical computation (579) of Hn.M athematicmllyj this mode is somewhat equivalent to the buckling mode1 - 11 - linder or in elongated prolate spheroids, if there is such az'il a:a 'n nxte cy ,lode in them-For lack of a better name, this mode rnn.y, therefore, be calledbpclrlz'ng- Surface anisotropy has also been included in some computations348) of the whole hysteresis curve. Formally they were carried out for(éiveral two- and three-di=ensional shapes, but since the magnetostatick' l ded the astual shape cannot really play any signl6cantnerr was not inc tt ,role. Beside its inclusion i'a the exchange ruonance modes, mentioned inS,èction 10.1) su'rface anisotropy has also been included in a recent study(580) of the unifoz.m mode.

1. 1-3.3 Prolatû s'.pàcrtétfMothing equivalent to eqn (11.1.4) has been designed for any ellipsoid otherrihan a sphere, which makes it dilcult to compute any of its properties.For the prolate spheroid there Ls some, but very limited, guidance 1omnnnlytic cazculations. It is known that the uniformly magnetized state isihe lowest-energy remanent state below a certain size, but only the lowerhound of eqn (10.5.41) caa be give,n for that size, and no reliable upperbound has ever been calculated. The nucleation has been proved to be bycoherent rotation or by curling. The possibility of a third mode ha: not

256

1:*e.11 ruled out, but it was shown that it may only eist for unpracticallylong and narrow prolate spheroids, in the region with a question mark iaFig. 9.2. For tke same surface anîsotror as in the sphere, its e-E'H on thenurling nucleation ield is (553) to modi'f.r only theeue of q in eqn (9.2.31).

There aze only two numerical studie (531, 581) of prolate spheroids,and only for an mspect ratio of 2:1. ney used time-consuming methods tocompu* the hysteresis curve for several particular radii, with and withcutanisotropy- The nucle-ation Geld increased (581) by exactly 2A%/Ms from$ts value for Jtez = 0 but tàe coercivity incren.qd!d by less than 2I%(Ms-

The smallest semi-major >=' s, c, t'I'iC.II is reported aq Clleah = 0.04,whicà is probably mpltnt to be C/(4<rc2Ms2) = 0.04. Ia the notation of On(9.2.20), the semi-minor nlv5s is S = 1.0, which is in the re#on of coherentrotation in Fig. 9.2. Eor llik radius) there was one jump at nucleation fzoma podtive to a negadve saturation, and st wms not clea.r visually whethercurling was very sûght or nonefstent' (5811- The next size they kied was0.02 in those units, which should me= S = 1-4, well above the transition tothe curling mode in Fig. 9.2. A wemdevdoped curllng strudnre wms folmdto nudeate for this size. but no attempt was made to check the change-ove.r size- Neither was there any at-mpt to compare the computed curlingnudeation felds with the analytic Gpression, not even in order to checkthe accuracy of the computatlons.

The curûng confguration ai this Ae, whic.h probably corresponds toS = 1.4, wa.j (531, 581) qui/ similar to that of a sphere. The hrteresiscurve consisted of a continuous càznge from nucleation dou to a certainfeld, at whc.lz there wms a jump a11 the way to t:e negative saturation.For a certna'm larger radius, this jump brought the magnetization into axrling state wltich was a mirror image of the structure fzom whic,h thej=p started. More complex behavioars were found (531, 581) for still largerpartides, induding some fo= of two domains with a complex watl betwenthem. However) all these cmses were studed for one spexc z'adius eaclbwit,k no at-mpt to follow the t'rxnRitîon from one cxase 'kh =othcr. Neitherwas there any attempt to follow the drastic and qualitative change of thehysteresis bdween a sphere and the particular aspect ratio of 2:1. R isprobably impossîble to do any more by this method. whic.h is very wmstefulin computer time, but one would still expect such reports to #ve some ide.aof the estlm ated accuracy, or convergence criterion, or at lplmt how thecomputer wms kept from jumpizg from one possible solution to another forsome of the rnzlii- The published arkcles do not even mention the relationbetween the pvticle K*,> and the dtqcretizatlon size used in tYs work.

NUMEMCAL MICROMAGNETICS

11.3.4 nïzz FilmsSome computations try to account for the exp'm-rnental hysteresis curvesof thin %lms, and relate the.m to some measurable properties. One wayto do it is to condder the fzlm as a collection of non-interacting particles.

COMPWATIONAL RESULTS 257

By pretending that the 'wltole hysteresis curve of each partide is knownta certain average is computed 'over the distribution of the particles. Suchcomputations, e.g. (5821, sometimes reproduce the meastred properties,at least qualitatively. Of course, this process is not limited to thin flms,and has been used for other systems of particles, including (2144 models forinteracting particles. ln this context it is nlscï worth noting a computationalscheme (5834 for fnding the whole magnetization confguration in a thinflm from experimenul data of Lorentz microscopy, and of the measurementof the magnetic feld pattern outside the 61m.

Other computations are concmrned with the domains in thin flms, andare done on a rough scale which cannot take into account the walls betweenthe domains. For this particular purpose, it is convenient to ex-pand themagnetization in a Fourier series (584!,

M(r) = J'l pkefk'f , (11.3.5)

k

where k has certain discrete values in the zv-plane, such ms a.n z-componentof the form znr/.f?x for integral values of .?z. The coeEcients gk may bcconstants or fanctions of z, and in either case the solution of the potentialproblem fxom the dferential equations of section 6.1, by expanding thepotential in a similar Fourier series, is vezy much simplifed. The diEcultyis that it is usually impossible to flt the expansion in eqn (11.3.5) with theconstraint of eqn (7.1.7). For this remson, this method could actually beused only when the walls were taken to be step functions (.%41, or in somesimilar applicatioas reviewed in (2884:

A similar technique has also been used in many other computationswhich impose a false periodicity, in order to use the fast Fourier transformswhich reduce (550) the computation time by a very large faztor. It hasbeen argued (550) that this approfmation is justifed for a two-dimensionalsystem, because 'the efective range of demagnetizing feld is comparableto the f1m thiftkness'. This argument sounds quite convincing, but it wouldhave been more convincing if there wms any casc for which a computationusing the fast Fourier transform was compared quantitatively with a morerigorous computation of the same case. In a mriation (542) of the method,the potential is expanded in a Fourier series, but the magnetization isnot. The ex-pansion is used to eliminate the potential outside, Uouty andformulate the whole problem within the (infnite) ferromagnetic 6)m, withthe boundary conditions expressed as an integral over the upper and lowersurfaces of the flm. But even this formulation is actually applicable only toa perîodîe domain structure, for which the potential is really periodic. Orat least it has only been used for such a periodic confguration, in a crude-mesEed computation (542) of domains, that oversimplifœ the structure ofthe walls separating those domains. ln a certain study (585) of very thin

258

fllm s, it was found adequate to aegled the magnetostatic energy altogether,except for restricting the magnetization to be in the plane of the film.

Another method, which is also confined to two dimensions onlù takesadvantzage of the upper bound to the magnetostatic enerr in eqa (7.3.46).In prindple, the magnetostatic energy term caa be replaced by this upperbound, and the total enerr czm be minimized with respect to both Mand A. Mà-niml'zation with respect to A makes the upper bound convergetowards the true magnetostatic energy. Therefore, this mimsrnleatton leadsto the true minimal enerr confguration, while A becomes the true vectorpotentiaz of the problem. The advantage of this technique (fzst suggctedin (586) , but never carried out by these authors for any particular cmsel'is that the sk-fold integraz for the magnetostatic enera is replaced bya three-fold one, which should reduce the computation time enormously.Increasing the number of variables from the two tdependent componentsof M to the fve components of both M and A Ls a smazl price to pay for thislocalization of the problem, which does not call for evaluating interactionsamong dferent discretization points. The disadvantage, azready discussedin section 7.3.4, is that eqn (7.3.46) contains a.c ttegral over the wholespace, which in practice means c-arrying the ttegration over a much laigervolume than the sample, thus increasing the number of grid points farbeyond those used in more conventional methods. This dllcalty makesthis method impractical for almœt acy three-dimensional problem. IIl twodimensions, however, the htegral over the oater space may be evaluated bya conformal mapping of it, mahng this method pcactiol and convenient. Ithms thus been used, for example, in the study (587) of the efect of Gchangecoupling across grain boundaries of the nucleation feld of a 6lm.

A particularly popular computationaz method subdivides the :lm intoeither two-dimensional hexagons or three-dimensionaz hexagonal columns.The hexagonal 'grains' are mssumed to be somewhat separated fzom eachother, so that the exchaage coupling between them ks smaller than it Ls in acontinuous flm- H practice it actually means that the expression (588) forthe exchange energy between neighbouring grains hnm the sume functionazform as described in section 11.1 for subdividing the sample into prisms orcubes- Of course, a hexagon has more neighbours to interact with than asquare. But besides this dxerence in the summation, the only dseerenceis that the numerical value of the exchange constant, C, is taken to besomewhere between 0 and the experimentaz vazue for a continuous 61m.The actual mlue fot this efective C is often picked at rndomj althoughit is possible L5891 to estimate it from memsurements of the domain wallenergp The coeEcients for the tensor describing the magnetostatic enerorhave been emluated Annlytically (5901 by integraing the surface charge,on the faces of the hexagons, in a similar way to that used for the prismsdiscussed in section 11.1, but many computations use an approfmation forthose coeëcients.

NMRICM MCRONIAGNETICS

COAUTATIONAL RESWTS 259

This modd was used for two- and throdl-men-qional computations of:0th the statics and the dynamics of magnetization patteras that do notinvolve the fne detna-lK of the walls between the domains. These include,/or evnmplez the magndization ripple, or the formation and realrangement.T

if domins and other covgurations, and their efed on the fhll hysteresispurve for the whole 111n1, ms well as a dmulation of the magnetic recordingjroc-. The detzu-lK of these computatlons, and all their results, are fully8, esMbed in (5881. Thts detailed description, however, does not give anyYformation on the choice of the discretizaton sîzea ox of the con-gencecriterion for tke distribution ia spacm It does specify tàat the step size infïme was chœen so that the mxvimum relative chaage of the magneeationrin that step was kept at approximately 10-4. lt is nlgn mendonM tdhat aï'$ - - - - '-

. .Lhaaetization consguration was accepted as a rnxnzmum oaly aaer trying.'jIqio' add to tt small random perturbations, and checkiug that it evolvH bnrlrïyto the initially obtained. confguration. Tt is a aice check on the validity,èf'ihe miaimization processl which should be adopted by other workers ac

# ell, bemuse it guaraxtt- arainst connerging into a saddle point in the'énerr mxnifold. lt does not guarantee, however, against the computations1 -- -

. .ienverogiato a high-eaergy rntnlrnum when a lower minimum is avdilable,V fn'nlly since thce computahons are not made to starq fzom a well-pe .

sûeEned nucleation- The use of this subdivision iato hengons. coatinues,'è..g. izï studying (591: the efect of va.ia bouudarie or of (592J a r:mdomln-lnotropy. It was also eended (593) to elongate àe-gons.

Some computations addras the mxNctlzatîon confguration inside sucbA he-xagon (5942, or a one-dsmensîonal stacb-ng (551, 595, 596, 59% of m-rnx-lxvbengons. There are alsc computations of rvxrtltnptlar partidu made outbfthin ilrns (531, 59% 599, 600) 601, 6021, or a pcïr (6(k% 604, 605, 606) ofmc,ll rectaqgle, aad =ious other two-dimensional (6074 shapeas. A class byttr-mlf are plxna:r arrays of tbln 6lm particlu gzlz.kl, 60% 609, 610) for whichjhe computations are being compared with expezimental studie, and the'>.,' meat senmK to be quite good. The published articlœ, however, are notV te t for dzawing any conclusions.. equa ye

Such compuvtionsof rect ar particles used to be critiched by someworkers, because they ignored the insnite dema>etieg seld at the cor-'àûrs. nis dilculty wa solve by the demonstntion (544) that the coz'ne.rsl'Vve a negligible efect on the computed mavetization covguradon, ift lie subdivision is sAtmciently fne. It has adually been clm'med before (392)iàkat it takes onlv a deviadon of the oxdG of aa atomic size to reohœ that). *- -

.lTàâzllty by a rather small value, but the argumeat remaiae controwrmx'l-zàhe new approvî 2544) provu tkat it vms not even necasazy to go down to7 $ .?Jr.atomic ee, and subdivisions smaller than the exckange length of the'mxe were adequate for removing the divergence at the corners. Suchàlresult may se= strange, but an analytk model (611q gave it a plausiblepliyical eolzmation, not only for a twodimenskonal corner but also for the

260

thrn-q dimensional one dkquu-q.qdvl in the nex't section.Tf it were pMdble to saturate a plxnztr square, tîe enerr needed to keep

it saturated would have been isotropic (see the araments in secuon 6.3.1)-The very slight deviations fzom saturation, however, do depend on thedirection, thus causing a tcovgurational' anisotropy ixt squarœ, as found(612) b0th in computations and in erperiments on some thin flms.

11.3.5 Prism

NUMRICAL MICROMAGNETIG

hs mentione in section 10.5.3, Brown's r'fundxmeatal theorem' does nothold for a cube, aad there is no m'vp below whicà the r-anent state of acube vill be the uniformly mMnetized one. Actlmlly, this condusion couldhave ben drawn d-l-ctly from eqns (8.3.37) and (8.3.38), whic.h inclcde, inprinciple, a11 the minimnm enera states. lf tke hlnsfoz.m state, 'rzza = 1 andmz = mv = 0, is substituted in these equations, it is seen tîat ther caube fulflled only if S'z = Hv = 0. And these relations e.xn only be fulWlled5f either the body is au ellipsoid, or the applied fe'ld is not homogeneous.Eowever, this property of a cube was not seriously discussed lml;il somecomputadons (6134 revealed the equilibrium states of such a cube, andmade the problem quantitative.

Before discussing th- result, a semantic point needs to be clarled.Woen Brown (520) looked into ihe nucleation in an l'M4nitely long prism,he noted that all ihe possible eigenfunctîons could be arranged in groups,according to the symmetry clmss of the Omponents zrz. and n6. This dms-Xcation waq tho extended (2912 for the case of a rectangular prism,-a K z f % -b S y K à, with z exiending all the way to l'n4nity. Inparticular, the nucleation mode for which m. is an even function in z andan odd function ia y, while mu is odd in z a'ad even in yb was #ven thename kurling', because it is basically made out of a ma&etization vectorwhich goes around the prism iû quasi-dzclœ. It is topologically the umestructure as Gat of the curling mode in a sphere, or in other ellipsoids- Themode for which m. is odd in m and even ï.a y, while mv is even in z andodd in y, lxks 1% the vedors describing the :ow out of a centre. lt wmsgiven the nxme fanticurling'l because all the above-mentioned symmetriesare opposlte to those of the curling mode-For some readers it may be easierto visualize this structure by following the equations for the componots ofmw ma , etc., as sped6ed ixï section 10.5.3. When t>e same mMnetiza-tion structura were rediscovered œs mssible minimal energy statu (613) inzexo applied Eeld, it was somehow felt necessary to #ve .them new namœ.One of the rp-œenns rnxy have been an attempt (613) to draw a distinctiveline between the sclassical' micromagneticsy and the new, numerical studies-The stated justifcation was (613) that the nxme tcurling' was 'œmmonlytaken to mean the reversal mode' and as such it did not ft ms a nn.me fora state. Therefore, the curling was renamed the Lvortex conf guration' axdthe anticqrling was renztmed the Towez' state. I never czmld âgure out why

COMPUTATIONAL RESULTS 261

two magnetization confgurations that look the sarne cannot be called bythe same name, but the new names have stuck in the meantime (576)) aadare commonly used by many.

' The zesult of these computations was that in zero applied feld thelowest-enera svte was that of the anticurling, or vortex, below a certainsize. This statement was formulated morc cautiously in (613) , because theaccuracy waz not really sulcieni for very small cubes, but the result wasconfrmed in the study (S16j described in section 10.5.3, and then in (545q.Of coursej the actual confguration in extremely small particles is of vez'y '

little interest, because the system will not stay there anmay. For a vezysmall size, the superparamagnetism of section 5.2 should take place.

ln this structurez the deviation 1om the eazy-es direction z is onlyfor the magnctization near tze corners of the cube. For a sulciently f nemeshv the magnetization is (5451 parallel to z, with negligible deviations, ata distance of one exchange length from a corner. With increasing cube size,the magnetization at the cube corners tilts hrther in the radial directionjbut the magnetization inside the cube is still not asected, so that thedecreMe in the average Ma is small. Above a certain size, the lowest-energystate becomes (6134 that of curllng, or vortex, with a sharp decrease in theaverage Mz. A detailed and complete phase diagram of the magnetizationstructures f or diferent anisotropy constants and a still larger size, is givenin (545), and some results for diferent prisms are reported in (614) .

lf a large Geld is applied in the z-direction, the minimal energy fower)con:guration shrinY, nxrnely the magnetization tilts more towards z, butit never saturates by completely closing this structure . Therefore, it wasconsidered unnecessary to go into the nucleation of chapter 9, and the RIIhysteresis curve was computed (613$ by applying a large feld, reducing it1then reversing ît, and minimizing thc total energy for each feld. There wazno attempt to check whether the coercivity thus obtasned depended on thevaiue of the initial, Csaturating' feld? or at least no such check was reported.

Of course, it is formally true that it is not absolutely necessary to lookinto the nucleation if there is no saturation, and a continuously evolvingmagnetization confguration art be computed. However, it is too risk'y atbest to use this approach. The non-linear diferential equations have an

'

enormous number of solutions, not all of which are minimal enerar states.These solutions belong to difereni braachc, wkich are intermixed togetherin the non-linear case, and can only be separated and resolved when theequations are lincnn-zcd. 1.n my mind, allowing the computer to decide onhow to stay on one branch, and at which point to jump to another branch,involves a too-optimistic view of the ability of the computer. I believe thatit is better to study ellipsoids, f or which the demagnetizing feld is betterdefned, at leazt as a frst stage, until the basic problems are understood.But even in problems in which line-artz' ation in order to fnd a nucleationmode can be avoided, or does not efst, it must be somehow introduced

262

anmay, as a check, bdore the solution cAn be considere meanimgful.The problem of whether the sharp cozaer of a cube (or simllla bodic)

bnn any physical meaning hms been higbly conkoversial for a long time. Itis settled now by the estemati'c studic (5M, 545, 565) tàat justf its useif 'the discretization is suldently fne. R is not clear, however, whether asuEciently fne discretization is indeed used in all the published studiœ, '

and there are other uncertainties there. Even if curved bodies may be toodccult to deal with in the computationsy it is possible for evnmple to putrounded bodies, wkth a Hown demagnething feld: at the cozaers, wiooutcomplicatinê 'tàe r%t of the calculation. Foz elongated prisms it is possibleto taper ofthe edges, whic,h will n.1M make them look more like m=y of thereal partides as %en in the eleceoa microscope. There may also be otherways to indude nudeation, but none of them was ever tried. It is mx-tnlybexuse most pYple are happy to get rid of the audeation problem, whichthey consider to l)e an unnecessary auisance- It is not. It is an import=tguide on where aad on which branch to start the computations. It is anvential part for thcse who wan.t thel computations to have a physiœmeaaing, aad to allow an insisllt on how to continue 1om there. It is anuisanœ only for thœe who want to compute sometldng fast enough forproentiqg at the next confprenœ, aad do not waat to be bothered by thenecpmm-ty to check the validity of thdr results.

Even in computations of a simple cube as dp-qrtrl-bed in the foregoiag, inwhich the partkle never satarates, magnetization co tioas do not justkeep evolving continuously-There aze stlll (613, 615, 616J œrtzu-n tswitnhz-ngmodes': and 1 do not seewhy they must be dl-eingukshed fzom the lclxquical'nucleation modes which are decribed i.a chapter 9. For nvxrnple, in verysmall cubes, the b>ic structure of the 'fower' state is mainul'ne duringthe reversal, but in order to switch, this :owe,.r hms to dose frst. For asmall feld applied in the negative directiony the tendency of the fower isto open (613) further: whic.h makœ it more diëcult to reverse. It tztkas astill more negative feld to make tNe structure suddenly close, and switchinto the direction of the feld. For larger cubes, this fower open until itsuddenly jumps (613) into the curling co -ons. In either case, tkere ksa clpumcut eaergy bxvzler, an.d aztually ît îs quîte obdous that there is nohysteuis without a barrier. Giving the proccs a diserent name, such as

ajump or a switcà, do% not change the hct that a well-defned nucleationproceq confned to a pnirticulr =dG has just been described. Moreover,this mode must be the frst one encountered when the êeld Ls changed,wkich means the feld fœ whicN the enerr barrier just fattens, ms in .Fig.9.1. Evading this issue, and lettiag the computer decide on the Jump, maylruzl to the correct result, but it is certainly not guaranteed to do so. . v

Some of the reported rœults (613) for the very small particles are notmuclz dœerent 1om those obtained by the Stoner-Wo%lfnrth model, de-sebed in section 5.4, for ellipsdds. A closer look (544, 616) revealed that

NUMEMCAL NHCROMAGNFWCS

COUUTATIONAL RESULTS 263

their switching is actually by another mode, which does not efst in ellip-idids, and which was given the name splaying mode. lt should be helpfulitt include this mode in computations because the closeness to the knowncoherent rotation mode may be taken as some sort of a check of the com-

Ejuter program. The main problems, and the maîn dxcalties, however, arein the study of the bigger particles, for which there are sometimes tratherqèmplicated magnetization con:gurations' (6131, and much more care mustbe exercised before determining the jump into or out of such states., This necessity to be very carefal is even stronger in studies of elongated

trprisms made of little cubes, thoroughly reviewed i.xt (61$, as well a,s in theother cmses mentioned in that review. It also applies to other computationspf bodies with sharp corners, such as two interacting cubes (618, 6191, andthe cylinders which are discussed in the next sectiom There have also beenikore recent computations of elongated prisms, and indeed it seems (554):that they have not ben done carefully, and therefore lead to wrong results.r Knowing the nucleation feld always helps in removing serious mistakes,but it should be noted that it may not always be suëcient. For nvp.rn ple, inkhe computations of a sphere, discussed in section 11.3.2, only the be#nnjngof curliag eltn be checked against the analytic result for the nucleation feld.After tlzis start, the structure changes continuously, till a certain feld Lsreached at which there is a jump, and the curling confgulation is replacedb. y something else. For the location of the latter jump there 'is no analyticjuide, and computations in that vicinity need the same care which is neededlbr the ftrst jump în a cube or a prism. Of course, a program that reproducescorrectly the Erst jump is more likely to be reliable for computing the secondone as well, but one r-n.n neverbe sure. It should be better to havesomethinganalogo'as to the nucleation theory, which would determine the beginningpfa new mode, even when it starts from a complex confguration.' Iu prindple, the nucleation problem need not be defned (as in chapter9. ) in terms of a deviation starting from the saturated state. It is the onlyçase which has been studied in detail so far, but it Ls not the only possiblty.T. . here were actually some initial attempts at of a crude analytic treatment4f other cases too. In one case, the hystercis curve started (3432 by rotation6' f the magnetization along the Stoner-Wohlfarth curve: and jumping toèurling from there. There was no search for other modes, a=d the processwms not very diferent from those of chapter 9, because the Jump startedibm a uniformly-magnetized state, even if it was not the satuzated state.Jn another case (304), stabitity was checked by considerîng small deviationsfrom one-dimensionat maaetîzation structures. Such structuzes werefound

. to be always unstable, and to just collapse, so that it was not necessary tolétudy the details of the collapse.

There îs no special dimculty, however, in developing a more general the-ory, of a tnucleation; from a non-uniform magnetization state, say Mc(r)-U' ne way Ls to add a small perturbation, dMz(r), so that both Mo(r) and

264

Mc(r) + wMl($ are soludons of Brown's svtic equations (8.3.40) with theappropriate bcundary ctmditiona. By substituthg bnth Mn and M0 + Mtin thee equations, and leaving out evezy tmrm wlkich js higher tka.a thefrst order i.n e, it is pceble to obtain a set of linear diArential equationswith boundary conditions 4or determizdng Mz(r). It is then possible, iaprindple, to look for the whole eigenvalue spectrum of these equations,in the same way ms ia scme cues in Gapter 9. The Eeld vzue at wkichanother mode <11 start to Enudeate' wi!l then be deterrn-xmed by the srst-enovzntered eigeneue- Amotàer way, s-tudied (620) în more deta'il, Ls tostart from the expression for the energy, work out the fa'st vadation thatgives the eqnill-brium sta*s as in the derimtion of Brown's equations, butproceed also to tke second variation, whic,h determines the stability of tàatequilibrium. A jump becomes possible when the second variation vanishes,wizic.k l<mz!s to tàe sxme diferential muations. A matrix notation (612J caahelp solve the linenriqed muations. H either case, the kmowledge of theEeld at which the jump should occur can be used to guide the computer asto where to look for tkis jump or trandtion t,o another conEgmation.

The diEculty is not in wridng down the equationq but in solving them.Tàere is no case for whic.h tàe starting conâguration, before the jump, isknown in a closed form, œccept when it Ls the uiformly magnethed state.Q'heerefore, the only vay right now is to incorporate into the computationalprogram the search for a possible Tnucleation' of another magnekbmiionconfrzraticp. For this purpose it is ne to use suldently small Eeld-stem (or time-sleps where applicable), and to avoîd all sorts of short-cutsald approadmadons. Otherwise, the Omputation maylnst skip suc.h ajumpand continue elsewhere. Then, whe,n a jump Ls encountered, the provamshould go s''l dently back and rcstart tracing fzom Gere, using evem fnersteps, and a fner mesh. The coavergence criterion hms alrtuz!y been dis-cussed in suion 11.2, but it must be emphasized again àere that startingto compute in aaother âeld before the structure in a given feld is proptrlycompleted eAn lead to meaninglv resnlts. And) above a11) every provamshould contain a spech: searc.h for posëble rmddle points, with a check fortke possîbîlity of a jump there. If any of the published works contm'ned anyof these measures, tàey were not mentkoned in the publications.

The precautionstaken by (588) were already mentioned in sectioa 11-3,4.There a magneeation covguration was accepted as a minsm a2 energystate, only after trying to add to it small random perturbauons, an.d check-ing that it evolved back to tke initially oblmx'nM confgurauon. TMs methodcimost does what a nnvnerical nudeation theory should be doing, but theway R was designœl (or at least reported) just avoided C!J saddle points.Suc.h avoidance will not do, because 5.n the physical problem ms outlinedkere, a saddle poiat may be a tchance' for the meetHtion to escapeFom the branch it is onj to a lower-eaera one- The computcr should beprovxmm ed inemzi to stop at each f e1d value wkick does not answer the

NUMEMCAL MIGAOMAGNETICS

COMPUTATIONAL RESULTS 265

abovmmentioned citerion, look vound, and check whether the point mxy1M one 1om whicb it is possible 'to go to a lower-enero- confGration. lfit isy a Jump to the new branch should Vke place. lt may not be easyto formulate this SlooMng vound' i.u a programmer's language, but it rman

and shonld be done in any computation for whic,h the nucleation is notinvœtigxted =alyticatly. TMs generxllzation of the old nucleation theory,in a numerical form, is eqnally applicable for a sphere or for a cube, or

any other shape. If it % implemented, it will slow down the production ofresults, but it will produce only reliable ones. lt rnay l:e possible to skpthis stage, above a certn'-n pvticle size, if an unproved conjecture (621j,based on the results (615) of some cube computations, is found to .be gen-erally true. We suggestioa (621j is that to a frst-order approfmation thecurMg nucleaticn feld cf any regttlar body depends only on the nolnmo.Its dependence on shape is only through the demagnctivang fuor.

H is also essential to try to obtal 1om snch computadons more th=Just the numeric/ value for one particular rxv. It A-xlres a very small changein a program designed for a tperfectl pvticle, to mxlrn it n.lgn applicablefor fndinj out t:e eFect of defects. Thks efect was discussed in chapter 9:but theze lt was based more on guusœ th= on fnzre, and a bdte.r study isneeded. ln a recent computaticn the elœwt of mlrfnzte roughncs was studied(622) by removing certaln cubic elemeatts (or mn.lrin g them non-magnetic),in a particuhr pattam along a fromagnetic bar. 1.zl another dmulatîon(6232 (of a thin f1m) just one such cube was made non-maaetica either onthe surfaze or at the flm cenke. In both cases this drnuhtion çtî a,n imper-fecëon was found to make a signifcaat dxexence to the result, but b0thused rather crude subdivHons. It is possible to use the same technlque, ofcreating an inside fvcid' or a fscratch' on the surfaœ by removtg some ofthe little cnbe, for a deeper and more detailed simulation of the De Bloisexperiment..ln the xzne way, a dx-Ferent value of Jfz, or of any of the otherphyslrml parxmetersy can be exm-ly assigned for some part of the cube orpm-mm, and so on. Actuatly, cll the models for ezplaining the paradox ofBrowm which never rfmnhed any conclusive result,s by analytic ciculation,cau be vezy readily studied by snch a slight modifcation of the efstingcomputer propnms. It must be emphasized agnl-n that without Howingthe egect of imperfuionsj any comparison with eoeriment cf the compu-tational resnlts for ltd-t particles is meaningless and misleadtg.

No serious efort has ever been put into this sort of simulation, but itseems to be the onlk way of gaining some real phydcal information aboutthe tmle nature of the magnetization procœs in real particles, and a phys-ical indght into how to proceed from there. Hstead of such an attempt tosolve the real problem, the Dterature is jnst getting Ved up wlth results ofcomputations of m=y diferent aad uhcorrelated. e-q-qœ which involve dfer-emt a:d unspedfed arbitrary assurptions. '.F'.MK method cmm lp.ldzl nowhere,and the situation cannot improve a,s long as people hold tke mistaken idea

266

that such computations are in a Hi/erent class of a ïnew' micromagnetics;which should not be confused with the old, (classical' micromagnetics. Itis quite possible that such an efort to solve the problem of real particles.will run out of computer resources Yfore rsuzrh-tn g a size for which some-

, thing of jnterest may happen, as was the case with the attempt to compute. '

correctly the domain wall structure in thick flms. However, if this limit 1s..reached it will at lemst be known how far meaningful computations rnny bepushed.

There are other Xects which need a more serious consideration thanJis given to them. For ex=ple, eddy currents are known (6241 to be veryimportant for large, metnllic particlG. They cannot play an Mportant rolèin smazl particles or ver.y thin Slms, but a more ésczztïjcti'tœ estimate of.the limit to which they may be neglected is still missing. The problem ofa material in which two difereat phases are mixed together, known as ananocomposite magnet, hms not been solved either. In particular) it Ls notclear what boundary conditions should be used in the interface (see sec-tion 10.3). Some information cxn be obtained ftom numerical computatons(625, 626), bat there Ls no clear-cut theory. Also, computations of the enerrbarrier for a superparamaretic transition are still done (176, 196) 627, 628)separately. Jn prindple they should be combined with the computation ofthe statk hysteresis, tnblng 111t0 account the possibiliiy that thermi a#-tation may help the static jump, thus reducing the coercivity.

11.3.6 CklinderPractiYly a11 the discussion of the cubes and prisms in section 11.3.5applies also to the study of a inite circular cylinder. It is ltsted separatelyhere for two reasons. The frst one is that for this case of a fnite cylinderthere is an analytic proof (629) that the uniformly magnetized state r-qn

never be the lowest-enera state in zero feld. At some stage I tried to provethe opposite, 1om some upper and lower bounds as in section 10.5.1. lnthe evaluation of the lower bound 1 used eqn (7.3.43) with a certain vectorH'' instead of the V9 as written iato the equation here. I did not noticethat the vector H'' whicà 1 used could not be the gradient of a potential,because V x Hn was not zero. This mistake was poiated out in (6291, andI have already reported it in (5761, but I fnd it necessary to emphasize itagain.

The second reason is the need to mention a perturbation scheme 26304for calculating the deviations of the lfower) structure 1om the uniformmagnetization in the cyljmder. Dieerent analytic approimations are de-rived for a dat and for a,n elongated clinder, and the plotted spatial vazi-ations in both cases tunl out to be in fair agreement with those of thenumerici computation (526) for a cylinder. The subdivision of the latter isinto >5e.,$, and it seems to be a rather crude one. It is not clear fl'om thepresentation which are supposed to be more accurate, the analytic or the

NIMRICAL MICROMAGNBTICS

COAWATIONAL RESUZTS 267

numerical results. But the method is certainly unique, aad the attempt torepresent namerical resits in a.u analytic form should be encouraged. Ajimilar analytic approfmation was also GCXII (631) for a prism.' Even the most recent experiments on elongated c'ylinders, such as (632,d33J, go bvk for htemretation to the old theory of aa insnqtc cylindez'. It isuèrtzu'nly not n=ssatyto do sotespHallyfor the aaguhr dependence of thepucleation Eeld, for whicà the theory of a fnite ellipsoid (345) woald havebln better to apply. A particularly interesting experlrnent repohed (6341that the aagulaz- dependence of the switching feld had some of the featuresofthe curling mode, but was kdepeadent of the mw of the cylinder. H spitebf the discussion in (634), it should be obdous 1om Gapter 9 that a size-Ndependent mode aad the curling mode are mutually Gclusive, becauseike curling dœs work agalnst exchaagej whic.h vazies as S-2. Detnllq Hvekot bMn given, and it is quia possible that this size independence is ai œalt of uslg R in a region for which the f.rst te= of eqa (9.2.18) is mucklarger t11% the second term, or Kirnilarly in eqn (9.2.31), ms carl be seenin Fig. 1 of (392J. If it is a KZJ size indeppn/lencej it is a real challenge totEebrists to fnd, by analytk or aumedcal methods, what reversal mode ismeasured in this experlrnent.

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589. Muller, M- W. and lndeck, R. S. (1994). tntergranular e'xchange cou-pling, J. Appl. #?ù#:.j T5, 2289-90.

590. Eughes, G. F. (1983). Magnethauoa rev-mld in cobalt-phosphomzsElms, Z Apg #A&d., 54, 5306-13.

591. Schze:, T., Sehmidts, E. kn-, ndler, J. and Kronmîiller, H. (1993).Nucle-ation of reversed domains at grain Mundariu, J. Appl. .Pà#s., T3,6510-12.

592. Muram O., Saito, K., Nautani, Y. and Eayashi, N. (1993). Com-puter simulation of maaetization states in a maaetic thin îlm consbt-inj of magnetostaticallz coupled vxlns with randomly oriented cubicnntsotropy, J. An1. #?ù2M., 73, 6513-15.

,$93. Zhao, Y- and Bertrxm, E. N. (1995)- MHomagnetic modeling ofmag-netic anisokopy ia t-ured tbin 4lm media, J. Açpl. 'hp.s., 77, 6411-15.

594. Càaag, T. and Zhu, J.-G. (1994). Angulaz dependencû measurezmentof individual barium fqrrn-te recording particles near the sin#e domainsize, J. Appl. .Pà#s., 75, 5553-5.

595. UesaW Y., Nxlcettani, Y. and Eayashi, N. (1991). Micromagnetic cal-culadon of applied fdd efed on switching mechanism of a hexagonalplatelet particle, Japan. J. Apyî. Phys., 30, 2489-502-

596. Nxk-ettani, Y., Eayashi, N. and Uen.1rn., Y. (1991)- Computer simMla-tion of maaetization reversal mehxnlsms of haxgonal platdet particle: .

TEfec't of matedal parameters, Japan. J. Aypl. Pltys., 30, 2503-12.597. Uesalc, Y.j Nnkxtani, Y. and Hayuhi? N. (1993). Computation of

switc%lng felds of stacked magnetic hexagonal partide *1t,1t diferent

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364 RBFBRSNCBS

AUTHOR WDEX

The n'ambers tu square brvlrets are the rderence numbers. TEey are followed by thepa> numbers.

Abd-Elmeguid, M. M., (46) 58Abe. K., (574) 252Abeledo, C. R., (218) l0:Abolmxnn, tj., I1sl) 91.Abralmrnj C.$ (477) 217, 221, (478)

217, 2l8Abrahxmm, M, S., (45) 58Aclzer. O.j (4731 216, 254, (474) 217,

254Adxm, Gh., (285) 144, (584) 257Aiy, A. Ao (552) 249Afanas'ev. A. M., (5132 232Aharoai, A., (5) 19, (921 82y (95) 82z

89, (97) 82, (110) 88, 1146)90, 178, 187, 255, (161q 91, 1082I170J 94, 95, (173) 95, (1742 95,(177) 95. (178) 95, I180J 95,1K, 225, 2309 (227) 102, 103,105, (244) 119, (253) 125, 225,230, 231, 241-244, 25% 25% (270)136, 137, 14% 14% 173, 212, 220,222, 239, (271) 136, (273) 137, .(276) 137, (288) 149, 151, 15%1,54, 1.$8, 173, 240, 257, (290)152, (2911 154, 2371 260, (292)158, (295) 163-165, 1809 (309)166, 167, :.71 2322) 170, 171, (323)171: 172, 180, 244, 245, 250,(333) 180, (334) 180, 181, 221,(335) 180, :336) 180, (337) 1.80,244, 245, 2511 (344) 186. 203,(3451 186, 2671 (349) 187, 222,13501 187, 2=, (358) 194, (359)194, 200, 212, (360) 195, 197,200-202, 214) (3621 195, 19%(363q 200, 201, (365) 202, (373)204, 20% 212, (382) 204, 207,209, 21p, (392) 208-212, 233,236: 25% 2617 (415) 209, 210,(418) 210, (422) 211, (424) 211,(461) 2141 (462) 214, 230, 253,2549 (4635 214, (472) 216, 217,(47S) 217: 226, (477) 2171 227.1(4.871 218, 219, 221, (489) 220,

221, (490) 221 (491J 221, (501)223, 224, (502) 224, (503) 214,22% (506) 22% (50$ 230. 252,I5llq 231,, 232, (51.6q. 234. 235,261, (522) 237, (525) 241, (527)2419 24% 250, (540) 246, 247,254, (553) 249, 255, 256, (554)249, 263, (572) 252, (575) 253.254, (576) 253, 261: 266, 1577)254, (5791 255, (6211 265

Ah-tlea; E., (:16) 88Akira., T-, :183) 98Akoh? E., (183) 98Mdred, A. T., (3G) 44) (4% 58? 63Aluopoios, b. S., (,550) 249, 25?All=? G. A., (457) 213Mvarado, S. F., (56) 61Amemt, W. S.j (470g 2l6Anbo: E., 2377) 204Andetmm P. W-, (9) 32A'adrëz W., (533) 24% (534j 243Appel, W., (5M) 243Arajs, S-. 529) 46 .

1e., B. E., (49) 58, 59Arzott, A-, (65) 68. 81) (66) 68, (901

8l, (91) 81Arrott, A. S-, (501 58, 81, (1591 91,

:277) 1:$8, (3312 17% (458) 213,(463) 214, (585) 257

Aruztarkavalli, T., (222) 101Asselîn. P-, (586) 258Asti, G.. (164) 9l: (1.67J 91, (168J 92

Baberschke, K., (143) 90, 91Bvhmau, K-, (394) 208Bxri, J.œ.. (580) 255Bxalxz E., (233) 105Bagnérl, A-z (566) 251Rsaceng., u., (2ay zpsmaldwln, J. A. Jr., (428) 211, (429) 211,

(430j 2l1Baltenspcger, -W'.) (2721 137, :460) 214:

(461J 2l4Balucani, U., (751 76

306

Bssv-xndbarrm, 5. M.j (169J 92Baras, P., (542) 24% 257Barbaza, B.. (456) 213Bardx, D. L, (30J 46Barnaâh J-. (140) 89, (486) 2l8Baron, m B., 2147) 90Bartek L. C-, (lo5j 87, (l06q 87Bm-'m C. P., (198) 10% (220) 101, (367)

20% 203, (413) 209, 2l0Beardsley, 1. A.y 2583) 257Bodqer, J. A-, (219) 10lRvler, J. 1-, (405) 20B. :447) 213Beeby, J. L-, (30) 44Bey , m S., (60% 259Benaltt A. J.z (141) 89Benoit, A-, (451) 213, (4.56) 213Berger, A., (302) lM. 24&Bergboh, m, (153) 91Bergter, E., (153) 91Bœkow D. V., (228) 102, l05Bekowie, A. B., (206) 100. (445) 213Bea'tr.una H. N., (210) 100, (35:) 191..

(+43J 213. (444) 213, (560) 251.(s61) 251, (562) 251, (563) 251.(s64) 251, (588) 258, 259, 264:(s93) 25% :602) 259: :603) 259,(6101 25% :613) 260-263

Bsllms, 1, M. Lp (219) t01Birgeneau, R. J., (72) 76Bishop, J. E. L., (235) 108Bissell, P. R., (163) 91Bloomberg, D. S., (277) 138, (331) l78Blueo J. L., (327) 172, 244, (330) 173,

252, (546) 247Bwlker, P., (2032 100Boûeau, 171., (5421 247p 257, (565) 251,

262Bolduc, P. B., (82) 78, 79Bolzonit F., (l6TJ 91. (168J 92Bfvtaqioglo, O., 14:/) 21OBovier. C., (25) 44Braun, E. B.. (35% 191, (364) 202,

(460) 214 .

Brebeck, 0., (460) 214Brommer. P. B.. (94) 82Brott, K 1,., (238) l08Brown, E. A.. (10) 33Bxpwn, W. F. Jr., !1) 1, 110, 129, 130,

149, (103) 87, (145) 90, 135:137, 138, 141, 145, 164. 17% 178,181-183. (171) 94, (172) Q4,(255) 129. (266) 131, (268) 133,

AUTHOR EOM

138, (289) 150. 183. (301) 163,167. 238. (&4) 164. 263, (338)181, 182. (3411 184, 216, (353)18% (354) 188, (355) 188-193.(383) 206. (423) 211. (442) 213,(507) 225, 226, 229, (508) 226,228, 229, (520) 237, 260, (521)237

Broz, J. S., (460) 2l4Brug, J. A., (258) 130. (265) 130Bryaut, P., (2461 123, 124Buessem, W. Ru (3971 208Buiocchi, C. 7.j (451 58Burke, B. Ra (5522 249

Cabnxxk Ra (168) 92Cxln, W. C.? (4855 218, 221caz, M.t (441) 213, 259Cilawayà J., (32) 46C<lea, & lt., 198) 85I (99) 85Callen, 1I. B., (98) 85h :132) 8% (339) l82Capio, C. D., (402) 208Carey, lt., (53) 60, 61Carbon, L. Wv (238) 108Gmspers, W. J., (18) 34Cdinski, Z.: (159) 91Celottal m J., (55) 61., 172, (326) 1:2,

244, (3274 172, 244Gende.m Z. J., (528) 24% 25411 246nhxmherlim m M.k (232) 105Cbxmpion, E., (602) 259Clzang, C--m, (342) 186, (3524 187,

(459) 214, (488) 218, (494) 222Chaug, T., (451) 213, (5941 259Citautwmj m W., (163) 91, (196) 99z

265, (211) 100, (213) 100, (2251101, 10% 103, 105

Chapmau, J. N., (300) 163, (330) 172,252, (510) 23:

Charap, S. H., (491 5% 5% 7311 1:4Charles, S. W-, (203) 100Châtelain, A., (219) l01Chen, Du-xing, (258) l30Chen, J. P., (217) 10lCùen, W-s (5321 243, 244, 24.7, (6001

259, (61)9) 259Clzenz, L.-Z., :156) 91Cherkaoui, R, , (437) 213, (452) 213Chou, S. Y-, (5121 232 Y

Chow, c. K,, (432) 2l2lXnal, M., (1421 89, 90Coakley, K. J., (327J 172, 244

AUTHORWDEX 307

Cochranr J. F., (792 77, 9% (159) 91Coey, J. M- D., (200) 100, 22% (351J

18'rCo&ey, W. T., 1174 9% (176) 95c'oh 0, B. m , (26% 131Coho, M. S.? 1122) 88, 1125) 88Cooper, B. a., (1411 89Coopec P. V., (328) 173, (3292 l73Coxiovei, A., (285) 144, f584) 257Cozen, lk. L., (132) 89, (260J 130Cornwelk J. >n'., (31) 46Cowburn, m P., (612) 260, 264Crx-tlc, D- J-. 152) 60, 141, 163, 165,

(3?A 173: (329) 173Cregg, P- J., (1751 95Crespo, P-, (2s) 44Crew, D. c., (226) l02Czxm4vneyerx D. C-, (243J 1l6Czws, R. W.y (606) 259Crothem, D- S. F., (175) 95Ctlller, G. J., (428) 211

Dulberg, E. D., (229) 103Dale, B., (204) 1.00Daltoa N. W., (4; 14, 18Duu, H., (533) 24% (5%) 243Dxm-elst J. M., 1215J 1*

. . D-, T. P., (38) 48Daehton, J. M., (6055 259Davio, K.) (203) 100Davl, P. :F'., (385) 207de Biasi, m S., (199) l00De Blois, R- W-, 158) 6l, 14131 209,

210, 1414) 20% 2l0Dedezicàs, H.z 1:149) 2l3DedeHGs, P. H., (26J 44de Heer, W. A., (219) l0lde Jongh, L. J., (731 76, 77Deza Torre, E., (259) 130, 242, (262)

130, (263) :3c, 242, (481j 218,(482) 218, 221, (5381 245, (555)250

De-l V-cn-hlo, m M., Is431 247De-moktitov, S., (12! 33Deaony, S., (6332 267D4portes, J., (396! 208Desimone, A., (2472 123De Wxmo, m E., 1469) 2l6Dîetzich, H. B., (112) 88, 208Dietze, H. D., (425) 211, (427) 2llD-zrm-trov, D. A-, (332) :.78, 234.: (3484

187, 25,5

Fabian, K., :5452 247, 261, 262nlco, C. M., (138) *9, (13% 89Falicov', L- M.z (7:J 77Fharlez', A. Nu (571) 252Felamxnn , D., 161) 63Fea-nandez A- A. X.., (l*J 7.*Farz-az'i, J. M., (I50J 91Fertj Ao (331 46Fidler, J., (404) 208, (548) 248, (5871

258, (591) 25% (625) 266FMvet, F., (473) 21% 254, (4741 217,

254Flévewvmcoty F., 1473) 216: 25* (474)

217, 254Fige-l, H ., (193) 9% (230) 103Finea M. E., (186) 9% 99Fiorani, D., 2189) 99,l0D, (43.21 213,

(452j 2l3Fiscàer, P. B., 1512) 232

Diozme, G- F., (15% 9l, (457) 2l3Doemer, M. :F'.t 1129) 88Inanet, D. M,, (5l0J 230Döring, W., (275) 137Dormxnn , J. 1,., (18$ 99,100. (437)

21.% (452) 213Dove, D. B., 12613 l30Doyle, W. D., (132) 89Du, Y.-wo (221J l0lDuft-, K. J., (382 48Dumrenll , K.: (13j 33Dualop, D. J., (181) 97, (182) 97, 100

(5141 232, 233Darst, K.-D., (2415 116Duvaly E., (25) 44Dyson, F. J.. E401 52y 56-58, l09

Eagle, D. :F'., (4344 2l2Fadware, D. M,h (142) 89, 90Edwards, P. L., (421) 2llBgxmi, T., (2841 143Blsrzm,41'.o1n , 1., (178) 95, gl80) 95. 100,

225, 230, (462) 214, 230El-mlq M.. (213) J 100, (225) 101-103,

l05 ,

Engel, B. N., (138) 89, (139) 89Engemann, J-, (567: 25lEnrle, .F,.y (2294 103pnlrln , .it. Jv (514) 232, 233Rn'x. U. (:$8% 207Brdœ, Pw (15) 33Briclfson, R. P., 12% 44

308

Fl=her, m, (548J 248Fisher, M. E., (6C 6% l23Planders, P. J-, (15:) 91, (445) 2l3Forl>.r'-', F., (361) 195, 197-199Fowler, C. A. Jr-, (51.7) 236Frait, Z., (466) 216. (467) 2l6&aitovâ, D., (46% 2l6R-anse, J. J. M., (149) 91n'edkln, D. R= (210) 100, (441) 213,

259, 1455) 213, (529) 243, 248,253, 254, (530) 243, 247, (531)2431 248, 253, 254, 256, 25% (532)243, 244, 247, (581J 256, (598)259, (599) 259, r6ô0) 259, (6071259, (608) Q59, (6091 259

n'-rnan, A. J-, (:0) 77y 89R'ek, & H., (161) 9ly 108. (356) 189-

193, 200Diedmnnn, A., El58) 91h 216Friedman, N., (116) 88k'Yytp B. M-, (5171 236PURMG M-s (1282 88n'lcnaagay H., (626) 266nzkushlmw H., (536) 245F'ulmek. P. F., (379) 204

AUTHOR &EX

Griinberg, Pn Ll2) 33Grfmterg, P., (4861 218Gubuov, V. A., (24) 44.Gvggenheim, H. J., (402) 208Guo, G., (555) 250Guor Y.-M., (556) 250Gyorrà E. M., 576) 77, t23

Rxd?-ipxnxyis, G. C., (204) 1(D, (205)10% (217) l0l

Hagedora, B. F-, (236) l08Hxbn 9 A.? g251) 12sHall, E. L., (445j 2l3Hauy D,z (378) 204E'anedav K., Il14J 88, 208. (201j 100,

226, (202! 100, 2264 (39,% 208HnrrlMtb C. G., (3llJ 16% (318) 169I'Iartmxnn, U., (306) 1.6% (3071 165.

(519) 236, 15735 252Harvem R. t., (4s) 58Hathaway, K. B., (578) 255Eauser, H.y (280) 141, (281) 141s (3792

204Haynth-v, E., (1281 88Hayashi. N., (1352 89, (2992 163, 164,

240, l,44, (535) 245, (536) 245,(5474 247 (551.) 249. 25% (574)252, (592J 259, (595) 259,(5961 259, (5971 259, (618) 263,(619) 263

Hayashi, T., (183) 98Hebbert, m S:, (543) 247Hegedus. C. J., (259) 130, 242 -

Eeinricl, B., :502 58, 81, (79) 7% S0,(159) 91, (2771 135, (4631 214

Hellenthal, W., :87) 79Heller, P., (64) 68Helmans J- S., (2721 137. (460) 2l4Hempel, K- Ao (166) 91, (449) 2l3Eendrsksen, P. V% (203) 100E'mnlozb M., :13) 33Eernzt'n do, A., (251 44Eerdng, C., (21) 42. 44z (28) 45Herzzea-, G-, (:69) 92Heubelm-r, R., (195) 99Eiblo, Mo (121) 8% (1371 89Hicken, R. 5.b (1q0q 91N-deinger, H. R., :4(xj 208, (426) 211,

(480) Ql8Rirai, H-, (2404 ll5mrano, S., (547) 247Eirsch, A. A., (116) 88

Gadbois, J., (627) 265Gangop-Hhyay, S., (204, ) 100, (205) 1t%GaZ'CIG N., EQ5) 44Garcla-Arribas, A., (169) 92Gaunt, P., :191) 99, (197J 99, :2241 101Gaeau, F., :580) 255Geîrsng, B., (1441 90tn.rm dwm , y'., (s80) 255Geogheram L. J., (176) 95Gessinger, H-, (109) 88Gracomo, P., :267) l3lGiardius A. Ao E44) 58Gibson, G- A.. (4531 213Gi-en, A- A. v. d.z (184) 98Gipzerre, A.. (6331 267Gllu, m, .(550) 249, 257Givord, D., (396) 208, (633) 267Goedmhe, F., (492) 222Goldfarb, E- B., (1S51 98, (258) 130Gome, m D., (5524 249Goto. K.z (%8) 20% (389) 208GrMmxnn, U., (153) 9l, (154) 91Grahxm, C. D. Jr., (284) 143Grebensnhz-kov, Yu. B-, (620) 264Green, A., (319) 169GHGths, R. B ., (594 62

AUTHOR MEX

Ho, K.-Y., (156) 91Eoare, A., (163q 91Hogmnnn : F'., (1371 89Hofltman, H., (300) 16% (330) 173, 252Hofrrnan, R.. W., (85) 79, r86) 79, (1471

90Holstein, T-, (39) 5l, 52, l09Holz, A., (431) 211, 233Honda, S., (3901 208. (409) 209Hone, D. W., (71q 76, 77Hoole, H., (539) 246Hoon, S. R-, (57:1 252Hoper, J. H., (133) 89Hoselitz, .K.. gll3q 88s 91, 208Hosokawa, Y., (409) 209Hothersall, D. C., (3131 169, (3141 l69Hsieh, Y.-C.P 04831 2l8Huang, M., (380) 204Huber, E. E- Jr., (1221 88Hubert, A., (248) 124, 168, 169, (249)

124, 168, 169. 225, (495J 222,(5441 247, 259. 262, (545J 247,261, 262, (6162 262

Hughes, G- .F'., (590) 258Humpbrey, F. 4-, (324) 172, 244, 247,

(566j 25lHuysmans, G- X'. A., (433) 212

ldo, T., (3681 203. 204lglesias, O., (233) l05lida, S.s (207) 100Inaba, H-, (128) 88Indeck., R.- S., (58% 258Inoue, T., (535) 245, (536) 245Isaac, E- D., (531 60, 61Isbida, M.p (128) 88Isbii, Y.p (282) 141, (283) 141, 225,

(346) 186, 225, (376) 204, (377;204, (464) 2l4

Ivrmov, B. A., r5051 224Iwasaks S-I., (436) 2l2Iwaspka-, S-h-, (582) 257

Jacobs, 1. S= :198J 100, (367) 202, 203Jakubovic-sy J- P., (2531 :.25, 22% 230,

231, 241-244, 25% 253, (278) 141,(321) 170, 171, (323) 171, 172,18% 244, 245, 250, (334) 180,181, 221: (335j 180, (336) 180,(3371 180, 244, 245, 2511 (49:1221, :502) 224, (503) 224, 225,(5092 230, 252, (523) 239, (572)252, (575) 25% 254

Kadar, G., (259) 130. 242Kambersk#, V., 7511 91Kanai, Y., (626j 266Kaneko, M., (371j 203, 204Kannaz, K. R., (222) 101Kaplan, T. A., (771 77Kare, W., (2.79) 95, 100Ifauynma, T., (41lq 209Kawakatsu, H., (3151 169, 17:., (3171 l69Kelley, M- 1.r., (55) 61, 172Kench, J- R-, (124) 88Kez'ns D., (440) 2l3Ket'n, D. R, (441) 21:$, 259Kiz'a, T., (128) 88Kircbmayr. H. R.., (61) 63Kirachner, J ., (325) l72Kishimoto, M., (412J 209

. Kittql, C., (279) 14lKlabunde, K. J., (204) 100, (205) 100,

(217) 101Klein, R., (542) 24*6 257Klemaa, M., g1011 87Knappmann, S.? (144) 90Kneller, B. F-, (190q 99y 2l2Knowles, J. E., (369) 203, 213, (370) 203.

204, (372q 203, 204, (374) 204Koehler, T. R-, (441) 213, 259, (529) 243,

248, 253, 254, (530) 243, 247,(531J 243, 248, 253, 254, 256,259, (532J 243, 244, 247, (581j256, (598J 259, (5991 259: (600j25% (601) 259, (6071 259, (608)259. (6091 259

Kojima, H-, (114q 88, 208, (388j 208:(3891 208, (3911 208, (398) 208

Komëne, T., (623) 265Komoda, T., (128) 88Kondo, 1f., (240) 115Konishi, S., (409J 209, (5041 224Kooy, C-, 2387) 207

309

Jaualc, J. F., (308) 165Jatau, 2. A.t (481q 2:.8, (538) 245Jnes., D- C., (624) 266Jirlk, Z., (102) 87Jofe, 1-, (192J 99, (195) 99Jobnson, 1f. E., (129) 88Jolivet, J. P.j (437) 21.3Joliveq J.-P-, (452) 213Josepb, A. L, (256) 130, (257) 130,

(264J l30Juds J- H., (380) 204, (451) 213

31Q

Korvlr-x, J-, (1.79) 95, 1(i0Korenrnaa? 'V'-, 222) 44z 46Konln'x!.- m (298) z6a; 1Kom-o<t-! R. A., (567) 251Köster, Er., (109) 88Kxtx-lr.uj A., (204) 100Koduga, P. lt., (287J 148Kouvek J. S-? (43) 5% (89) 80, 81Krauq P. A., (512) 232Ifmbsj J. J-, (5781 255Kronmimer. H., (1084 88j (10:) 88, (241)

116. g3A) 208, (400) 208, (401)208, (404) 208, (426) 211, (54:8)248, (*7) 258, E5S11 25S

Krop, K.l (179) 95, 1G), (1934 99Kubq O., (3V) 20% 204K - G. U., (222) 101Kumx, J., (626) 2%Kuo. P. C., (375) 204Icm-hxmx ,r., gze4 88, s:tKus.mzla, Ta, (390) 208, /0% 209Kwq J., (76i 77, 123 .

Kyoho, M., (;.28) 88

AUTHOR PYEX

N.u, 1. P., (351) 187, (493) 222Livagey J., (452) 213Livlngslon, J. D., (220) 101, (40F) 208,

209Odder, C., (151) 91., (433J 212(mttîs, D. K., (229) l03Lu, D., (604) 259Lu, E.-x.# (221) l0lLubaw M.j (223) 101lmbotvlqy', E. &, (190) 99, 212, (366)

20% (435) 212Lyberatos, A., /96) 99j 265, (2T1) 100,

(212) 100j (225) 101-103, 105

Mccltrrie, m A., (4X) 208, (448) 213Mcpadym Lt (450) 213, (456) 213Masn, Po (13) 33Mx'zllyy D.y :45:1 21% (456) 213Mnll-knqmn, 7. C., (239) 112, (340) 18%

(434) 212y (443) 2:.% (444) 2î3Miozemoë A. P.j (1.11) 88Mnosuripur, M.. (330) 173. 252, (483)

218, (550) 24% 257) (5692 252MaayMn. E. A., (513) 232 .

Labarta. A., (2331 105 Martliztaa V. I., (408) 20SLaBonte, A. E., (3011 163, 167, 238, Martmek, G., (M1! 116

:310) 168-170, 238, 239, 243-245 Mxv-no-ra, S., (121) 88, (131) 89Labrane? M., (558) 251, (55$ 251 Kfzmhiyama. E-, (741 76Lagaris? :. &, :r4) 137 Mxqstnlo, C- V., (274) 13'6 2476) 217Eaizut, J. A.? (206) 100 A/Gf'hom J., (142) 8S, 90Lxm , 5., (4382 213 Matsom M., (22% l03taudauj L. D., (63) 63, 65j 67 Matt/mzcci, G., (5'5.1) 252Laog, Pu (26) 44 Maugîn, G. A., (104) 87Laxkxj Mn (5651 251, 262 Maye, Jœcph Mwaz'd, (42) 55Leaver. K. Dp (31.1) 169, (319) 16% Mayer, Maria Goeppez't, (42) 55

(5681 251 Mayergoyv, 1. D-, (552) MSLedcrrnau, M.j (4532 2131 (454) 21% Mazauric, M., (633) 261

(454 213, (634) 267 Mazjewakil A.z (505) 224Leej Cv M., (459) 214 Mendey E'. K, (306) l65Lee, B. L.z (821 78, 79 Mcrgcl, D.p :4841 218Lee, E. W., (2351 108 Mermln, N. D., (701 'r6, 77Iepeve'ry m Au (107) 87 Mernm , R. T., (6154 262, 265I-< R., (3%) 208 Micbijima, M., (1282 88ImIZk F., (245) 123 Mickiitz, H., (46) 58Lcviason) L. M., (223) 101 Middelhoek, S., (294) 163Izevinsem-n, II- 5., (402) 208 Middletony B- K, (393) 208Txwvy, P. M., (33) 46, (34) 46 Miedemao A. m) (731 76, 771,ew1, B., El17) 88, (120) 88 Miltat, J., (524) 239, 244, (557) 351,Lv-Mhtensêm-n, A . I., (24) 44 (5581 251, (559) 251, (565) 251,Liedtke. lt.. (417) 210 262, (611) 259Lahitz, B. M.) (63) 63, 65, 67 Minnajal N., (269) 136, 137: :296) 163,Linde-roth, S., (203) lX (361) 195, 197-199

AUTHOR HEX 3l1

)Mlv-zxmnnnl-, M., 1129) 88 Nunes, A. C., (216) l0lMitchelly R.. K.p (4061 208Mitsui, Y., :623) 265 O?Barr, R., (455) 213, (456) 21.3, (632)Mitui, T., (ll8J 88 267, (634j 267Mizjaj J-, (193) 99 Oda, E., (240) ll5Mizuno, T., (377j 204 O1De11, T. E-j (2861 147Moutroll, E. W., (69) 75z 76 Oelmann, A., (417) 2l0Mook, H. A.I (37) 47 Oepen, H. P., (144) 90, (302) 164, 248,Morelock, Cw R., (435) 2l2 (325) l72Mori, N-, Ll00j 85 O'Grady: K-, 1213q 100, (225) 101-103,Moriyap T.,

'

(1.4) 33, (16) 33 l05Morrish, A. E., 1201) 100. 226, (202) Ohkosb.iy M.z g4llq 209

l00p 226: 2255) l29 Olsonz A. L., (293) 163, l64Mlrup, S.a (187) 98, 100, (203) l00r Onishchenko, E- V., (513) 232

1214) l00p 257 Ozdshi; K.p (54) 61Moskowitzo R., (262) l30 Ono, Ea (128) 88Moyssides, P. G., (254) l26 Onoprieakoz L. G.I (395) 208Mrymsovh 0. N.r (244 44 Oredsont H. N.y (2931 163, l64Muccini, M., (573) 252 Orozcot E- B.j (456) 2l3Msllery K.-H.y (351) 187 Ortb.y Tb..z (163) 91Muller, M. W., (465J 214, (589) 258 Osbomy J. A., (242) ll6Mûller-pfezfer, S., (5371 245, 252 Otip J. O., (5701 252: 16061 259Mulligan, B.p (176) 95 Oucbij K., (436) 2l2Murata, O., (592J 259 Ozn.lcz- , M., (450) 213, (4542 213, (453)Myrtle, K., :159) 91 213, (456) 2l3

Nagai? E., (396) 208Nxlrxbayashi, T., (128) 88Nalcac, E., 1282) l4lNn.lexmura, E., (1834 98Nxknmurw Y., (582) 257Nxkxtaa H., 1504j 224Nn.lextanit Y-, (299) 163, 164, 240, 244,

(535) 245, (536) 245, 1551) 24%259, (592) 259, (595) 25% (596)25% (597) 259, (618) 263, (619)263

Nwtoli, C., (2452 123Néel, L., (6) 2l, 27; 28, 31Neem=, E., (415) 209, 21QNembacK Emj 1432J 212Neagebauer, C. A., g81) 78Newell, A. J., (6151 262: 265Newell, G. F., (69) 75. 76Niez, J. J., (5422 247: 257Nssuida.; K., (377) 204Noaku) J- B-, (65) 68, 8k? (91) 82.Nogués, M.p (437) 213: (4524 2l3Nolan) R. D., (113) 88, 9lp 208Noordermeerh A., (510) 230Nordstrôm? L., (26) 44.Nozièru: 5--P., (633) 267

Papacftkwzziou, v., (204) 7.00Paretip L., (162) 91Pattoâ, C. E., (1854 98Pelzl, J., (163) 91Perlov, C- u., (482) 218, 22lPet-zynski, R., (580) 255P<xnrhanyr S- >)., (526) 241, 248, 266,

1614) 261, (620) 264: L6301 266,(631) 267

Petersp A., :163) 91Pfeuty, P., (68) 69Pierce, D. T., (55) 6lp 1727 (326) 172,

244, (3272 172: 244Pinip M. Gw :75) 76Ploessl, R.p (300) 16% :330) 173, 252Pomerrmtz, M., (88) 79Pohm, A- M.t (605) 259Popma, T. J. A.t (151) 91Por, P. T., (510) 230Potapkov, N- A., (471 58Potter, H. H., (2) 3Prange, R. E., 122) 44, 46Prenq P., (437) 213, (4521 2l3Prima.kof, H., (395 51, 52, l09Pn-,a=, G. x., (578) 255Puchalsloz, I., B ., (l30j 89

312

Tbugb, B. W., (49) 58, 59

Qu>oh, 1I. T-, (1581 9lI 2l6

AUTHOR MEX

Qvbford, F. J., (578) 255p.'aa o, G. T., (150) 9l, (160) 91y (468)

21.6, (4701 216, (496) 223Rzu-lrbe.r, Yu. L., (5801 25&'Rxmstyck, 1f., (544) 247, 259, 262Raacourt, D. G-, (215) 7.00Rao, C. N. R., (222) 101Rathqaau, G. W., (149) 91Nxtnajev=, S., (539) 246Ratnxm, D. V., 397) 208Raver W., (544) 247, 259, 262, (545)

247, 261, 262) (616) 262Ravel, E.. (473) 216, 254, (474) 217,

254Rayl, M., (4b! 58Reale, C., (831 7% 79Rsxqidal, M., (324) 172, 244, 247Roveldt, M- T1z.. (510) 230Rettori, A., (75) 76Richvds, P. Mu (71) 76, 'r6 (1341 89Richter, H. G.. (112) 88, 208, (479) 2l7Rice, P., (570) 252Riedel, E., 23031 l64medi, P- C-, (62). 63meger, M.. (399) 208mslztozu S. A.: (441) 213, 259Rodbell. D. S., (220) 101Rodé, D., (210) 100Rodmvq, c., r13) 33Roos, R., (449) 213Roshko, R- M., (96) 82Ruigrok J. J. M., E510J 230Thwse-k, S. B., (606) 259Rram P. Jw (327) 172. 244

Sacchi, G-, :361) 195, 197-199S , K. A., (505) 224Saito, K., (5921 259R.*.!rx, C., (386) 207Sakuma; A., (234 44.Sakttrai, M., (155) 91Sakarai, Y., :54) 61Sakutaro, T., (2071 100S -

, C., (450) 213, (456) 213-. sxmao , G. A., (44) 58, g1:7) 87Sxmboagi, T-, (1lE) 88Sanders, S. C.. (606) 259Sako, M., (282) 141, (283) 141, 225,

(376) 204, (464) 2l4

Sato, T., (2404 l15Sawatzky, G. A-, (6B) 63Schabes, M. E., (525) 241, (613) 260-

26% $17) 263Scheiufeiu, M. Ru (55) 6:% 172, (326)

172, 244: 2327) 172, 244, (3301173, ,252, (546) 247

Smblgmxonj B., (M4) 130, (49% 223,(499) 223, 224

Schmid, H., (440J 21% (441) 213, 259Schmldts, E. F., (587) 25% (5911 259Schnekder, M., (537) 245, 252Scbmolz, A., I6lj 63rehBnezzberger, C.z (561 61Pmhwm, T., (5VJ 248, (58% 258, (591J

259, (625) 266Schtudq S. B., (1241 88Schaler, F., (420) 210rszmnnery ;. x., 1122 :.k%>hultz, S., (5% 61, gz140l 213, (441)

213, 259, (4m) 213, (453) 213,(454) 213, (455) 213, (456) 213,(598) 259, (6101 259, (6321 2671(634) 267

Scbnl'z, B.. (143) 90. 91Schwee, L. J-, (2971 163, (5431 247Sdzweninger, P-, (3l2J l69Scott, G. G., gl9) 35Searle, c. W., :41.0) 2o9Seeger, A., g1;.5) 88, (3031 164, (399) 208Segavas H., (1272 88, 91Self, W. B., (421) 2llPwmmye, D. J.1 (351) 187, (493) 222Selwood, P. W., (2182 10ISemglzettio J-, :25) ,t4Sbx, J. C., (381) 204Slzi, Y.-b-. g221j 101shllka- K.j (386) 207, k62z) 26sShlw, V. P., (580) 255SMmada. Y., (3912 208SA-marawa, K., 2386) 207Shlnjo, T., (188) 98Shirane, G., (r2) 76Shieido. H., (128) 88Atoltz, B. Vu (408) 209Sbtrikmam S., :152) 91, El6l) 9l, 108,

(223) 101, (304) 1641 263) (305)165. (3431 186, 263, (356) 189-193. 200. (360) 195, 197, 200-

. :02, 214, (515) 233. 236Shull, C. Go (37) 47Shur, Y. S., (408) 209

AUTHOR DYEX 313

Shyambumar, B. B.. (541) 246Sievtrt, J. D., (148) 91Sllva, T- 1., (57) 61Skomski, Ro (351) 187, (4931 222Slonczewsks J. C., 71) 33, (498) 223.

(500) 223Snu-t, J., g8) 3l, 86Smith, K F., (384) 207Srm- th, D. O., (122) 88Smith, N.; (V51 218, 221Smytt J- F-) (440) 213, (4411 213, 2599

(59s) 259, (61(1 259Sollis, P. M., (163) 91Soohoo, R. F., (347) l87Sorensen, C. M., (204) 10% (205) 100,

(217) I0lSpaia, R. 5., (130J 89Spratt, G- W. D., :618) 263Strmk-sewicz, A.. (505) 224stxnley, 'H. E., (77) 77Stapper, c. E., Jr-. (2s2) 125, 225, 229,

24lStennw, M. B., (35) 4*6 (36) 47Stephenson, A.y :381) 204St3ckel, D., :432) 2l2Stoner, E- C., :234) 10% 10-6 108, 1:.5Street, R., (226) :.02Subramaniazu, S., (539) 246Suhl, E., (246) 1M,124Suzuld, K., (316) l69Suakk S., (316) l69

Toulouse, G., (68) 69Thuble.g E., (799j 208'p'eve.s, D., (17) 33, El611 9l. 108. (305)

165, (343) 186, 263, (356) 189-19% 200, (515) 2331 2361 (517)236

Tronc, E., (:.89) 99,100, (214) 100, 257,(437) 213, (4524 213

Ttouilload, P., (5241 239, 244, (557) 25l'Ihzeba, A., (419) 210: (518J 236 .

Tsuuba'aj S., (315) 16% 1770 (37.7)16% (320) 170

Tsyznbal, E-. (12) 33Ttlrilli, G., (1621 91

Uedap M., :504) 224Ue=-kn.j Y., (299) 1K3, 164, 240, 244,

(5511 249, 259, (595) 259, (596)25% (597) 259. (618) 263. (619)263

Unguris, 5.$ :55) 61, l72, 2326) 172,244 (327) 172, 244

Usov, N. A., (526) 241, Q4% 266, (6144261, (620! 264, (629) 266, (630)266, (631) 267

Uyedw R., (207) l00

Valez'a, M- S., (571) 252Vnnmurer, C- E-, (206) 100von de Brnxlr, E. P.. (18) 34van Delden, M. TI. W- M.? 2510) 230vaa den Berg, H- A. M., (250) l24vaa de Woxde, B-, (60) 63vwo de Zaagy P. J., (510) 230V= Leeuwew A. A., (138) 89vaa Vleck, J. E., ;71 23Varma, M. L9'., (861 79Veerman, J., g149) 91Velîœsku, M-, g61) 63V - A., (542) 247, 25?Viau, G., (473) 2161 254, (474) 217, 254Victorw m 11., (194) 99Violet, C- E-, (82) 78, 79Viscian, 1., (1364 89Voigt, C., (148) 91, 2449) 213Voltxlram, P. A-, (274) 137, (476) 217von Baeye-r, H. C., (20J 35Vos, M. J., (238) 108

wvquxnt, >*., (633) 267Wua, N., (2071 100Wue, R.. H.j (416) 210

Tx'kxb aabi, M., (1l9q 88Tnmagawa, N., (208) 100Tan-yx, T., (208) 100'raq4ald, A.) (207) 100Tawn, R. A., (32) 46Tebble, E.. S.. (52) 6t)y 141, 163, 165Testa, A. M., (4371 2l3Tlzeilet J., (5671 251q'iitiadlle, A., (456) 213, (524) 239, 244,

(:651 251, 262, (6112 259Thiele, A- A., (586j 258Tlmmpxn, A- M-, (300J 163Tognetti, V., (75) 76TomM, D.. (6llq 259.Tomltw 11., (74) 76Tomlinxn, S. L.y (571) 252Tonegtuzo, F., (474) 217: 254Tonomura, A-. (549j 249'Ibnomura, E-, (54) 61Torok, E. J., (123) 88, (293) 163, l64

3i4

Wagner, D., g3) 7, 12Waaerl H., F(( 76, 77W=1=- , 1t., (4121 299Wakui, J., (433) 212Waldron, J. T., (175) 95Walker, L. R-t (471) 2l6Wang, C. S., (22) 44, 46Waring, m K., (209J 100Watsen, J. K., (297) l63Wayae, R. C., (1074 87Wecber, K., (s39) 246Wd, M. S .h (512) 232Weissj G. P., (122)Weiss, J. A., (457! 213Welland, M. E., (612) 260, 264Weller, D., (57) 61Wells, S., (203) 100Wendbausen, P. A. P., (351) 187Wcrnsdorfer, W., (4511 213, (456) 213White, R. M., (229) 103Wickstead, A. W., (1751 95Wiedmasa, M. H., 738) 89, (139) 89Wijnj H. P. J., (81 31, 86 '

Willinms, C. M., (126) 88Willinms, G., (96) 82W-tzinmq, M. L-, (601! 259WiHmoreh L. E., (4484 213Wilson, m H., (43) 58Wirtit, S., (439) 213Wirtz, G. P., (1861 98, 99Wohlfartit: B- P., (31) 46. (93) 82,

(2344 105, 107, 108, 1:.5Wolfrnm, 1%-t (469) 216Wojtowicz, P. J., (451 58Wolf, W. P., (265) 130Wood, D. W., (4) l4, 18Wu, c. Y.y (158) 9l, 2l6wu, J., (221) l01Wu, R.-Q., (80) 11, 89WulGekel, W., (144) 90Wysîp G. M., (332) 178, 234, (348) 187:

2S5

Xiong, X.-Y.: (156) 91Xu, M.-x., (221) 101Xue, R.-b., (221) l01

Yaegasbi, S.j (127J Vj 91Yafet , Y., (76) 7T, 123Yxmadas H.s (51) 59Yan, Y. D., :263) 130, 242Yang, B., 2600) 259Yaag, J.-S., (352) 187, (459) 214, (494)

222Yang, M. H., (465) 214Yang, Z., (378) 204Yatsuya, S., (183) 98Yee, D.j (440) 213Yelon, A.h (158) 91, 216Yeung, 1.p (96) 82Yokoynmw H., (368) 203, 204Yu, Z.-C., (2161 101Yuxn , S. W., (5601 251, (5611 251,

(562) 251, (563! 251, (564) 251,(603) 259, (610) 259

Zelenr', M., (102) 87Zener, R., (26j 44Zitang, S., (33) 46, (34) 46Z'hx, Y., (593) 259Zhzso J., :156) 91Zhu, J.-G., (2381 108, (451) 213, (556)

250: (588) 259, 264, (5945 259,(604) 259, (622) 265

Zijlstra, E., (403) 208, 20% (446) 213Zimau., 7. M., (41) 53j 57, 76Zirnvnprvnx'nrt , G.: (1651 91, (166) 9l,

(237) 108Zinn, w., (12) 33, 2844 78t (537) 245,

252iukrowski, J., (179) 95, 100zuppero, A. C., (85) 79Zweck, J., (300) :63

AUTHOR DYEX

SUBJECT INDEX

alumitez 212amoxphous materials, 82, 89aagular momeatump 8, 1Oa 35anjsotropy, 26p 83p 84, 87, 94, 97, 109,

. 125, 181: 189: 204, 220, 253,254, 256, 26O

a'rtifdal, 89constpmt, 91., 10:1., 132, 140, 7.44, 7.70,

1719 208, 21,1, 217, 218, 221,249, 251, 261, 265

cvbict' 86, 87, 89, 95. 108, 115, 141,180, 186, 204, 215: 226, 228,230, 234, 252, 255

distributioa, 91, 9 2easy axis, 85-87, 89-92, 96 p 105, 107,

108, 113, 115, 116, 121, 138,141, 160, 203, 21.5: 226, 231,236, 253, 26l

easy plane, 86eaergy, 33, 84h 85, 87, 105, 115, 120-

123, 125, 133, 139, 144, 145,157, 159: 160, 167-171, 174-176, 187, 196: 207, 219, 226,231, 234, 243

hard axss, 86iaduced, 88, 89magaetocrystalliae, 84, 8% 9Oa 105,

115, 116, 189r=dom, 89, 91, 108, 259rovtable, 88shape, 88z 95, 105, 115, 116, 125, 189,

2O4- suzface, 89-91: 100, 138, 145, :7.76,

178, 180: 187, 200, 216, 217,222, 226, 234, 249, 255, 256

uaiaxial, 85, 88r 89p 92, 94, 99, 105,108, 115: 116, 138, 141, 157,160, 180, 186, 203, 215, 219-221, 226, 228, 230, 252, 255

anticursing mode, 260: 261aatife-rromagnet, 20-24: 26-30, 32, 33,

42-44, 59, 63z 7O, 76, 78, 79,99, 218

Arrott plots, 62, 80-82

Bal-ker:ilozv, 206: 207Bitter pattera, 60

Bloch law, 57-59blockiag tempcature, Tz , 98Bohr maaetoa, Jzs, l2, 35, 45, 48r 63Boltr's electroa orbit, 7-10Bohr-va.a Izeeuwer Geo-m 6 9Boltvnxnn 's coastantz kB? 6, 7l, 83Brillouia functloaz 14, l7z 22z 24z 25, 28,

8Op 83, 97Brillouia zoaej 53, 57, 59Browa's ..t, lt 110, 114, 129, 131, l32Browa's equatioas, 173, 178-181, 183,

186, 187, 189-191, 194: 198,200: 205, 217, 219, 220, 223,225: 264

Brown's ftmdameatal theoremj 225,236, 26O

Browa's paradox, 204-211, 236, 265Brown's upper & lowe,r bouads, 149,

226: 253: 258Brown-Morrish theorcm, l29Bsn =ce Brillouia fuactbabuckliag mode, 200-203, 205, 255

ICHaNNMACIA, sec TMMCCo, sez cobaltcobalt, 42-46, 96z 125, 135, 212: 222,

231, 253coercive force, sez coerdvitycoercivity paradox, sr.r Brown's pazw

doxcoercivity, Hcb 2, % 88, 99) 107, 108,

205-208, 211-213, 217, 218:254-256, 261, 266

CoFea O41 222cohereat rotatioa modey 188, 189, 191.-

194, 196-203: 205, 206, 216,225: 231, 254-256, 263

compeasatioa poiat, 30complex coajugate, 37p 39, 40coaductioa elcxctrons, 36, 42, ,t14: 48, Wcopper, 44-46correlatioa leagthz 62, 69citical expoaeats, 66-69, 80-82: 84,

124, 125critical iadex, Jcc critical expoaeatscritical resoa, 34, 66, 69; 8l, 89

316

cryatp.llln c imperfections, Jea imperfec-tions

Cu, 4e.e copperOku'le

law, 15temperature, Te, 3-5, l0, 22-24, 26,

29-31, 34& 441 58y 62-68, 70z76z 78-81, &i, 85, 101, 7.36

forrlmagnetic, 29, 30, 32pxarnagnetic, 29t 32, 58

Curie point, aee Curie, temperatureCude-Wdss law', 22-25, :$0, 66curlingco dom 217, 256, 2*-263curllg mode, 189-.$94, 196, 197, 199-

206, 212-214, 2164 225, 224-231, 243, 253-2561 260, 263,26% 267

delocaliyed, Je.4. îtinerant electronsdemAtrneklez-xà-s on, demagrzetO 2, 128,t

141, 210, 2l8factor, 114-116, 128-130, 132, 187,

188, 219, 220, 231, 265yssllx-ttîc 130ma>etometric 130

?'

âeld, 114, 116. 118, 122, 128-.130,142, 143, 186, 181, 204, 210,211: 232, 233, 236, 257, 2$9,261, 262 r

diamxgnet, zI1%m= retirn, 1, 7, 8, 10,11I 15 .

dipole, 4, 77, 79, 109, 111. 123, 142, :43,145, 213, 234, 240

moment 1, l2, 87y 132, l42dislocation, 88z 208distrîbutlon

augles, l08aniwtropy, se,t anisotropyBolem xnn 92dislocatlons, 88enerr barriers, :02-104lme-tîon, 101, l03

gnmma, 103-105paztides, 257relaxation tîme, 101s 102, 104size, 98-102, 2l3wîdth, l03

domain wall, 6c, 99, 120, 121, 123, 12$.127, 133, 137, 1.38, 140, l41j156, 157, 179, 180, 207. 208,210, 217, 219, 221, 223-225,231, 238, 241, 244, 245, 247,248, 250-253, 256-2$9, 266

SUBJECT EfDEX

Block 157-:63, 165, 168-172, 250cw--tie, 163, 164, 169, 2$0, 251Laudau and Lifsbitz, 133, 138, 143,

157, 161, 164, 171, 172,, 174,250

mx-cql 222, 224Nlel. 1$7-165, 172, 180, 250, 25l

domainsconf guratiom 61y 1201 128, 141, 173,

225znaaeticz 5, 20, ;7, 47, 60-63, 66, 69,

81, 82, 85, 87, 89, 9l) 93) 99,116, 118, 120-125, 127, 128,136, 138, 158, 171, 207-212,225, 229-231, 236. 238, 245,247, 258, 254, 256, 257, 259

Wes, dee Weissl domainsDöring 'rnaq.q , 224

S&-nmtein-de Enx.q experiment, 35electron spin rœonaace, 79entmps 65Buse, 21 -

excbxnge, 112, 16, 17, 20-22, 28I 33, 36I43-46, 62, 76-79, 82-85, 89,120, 124, 133. 137, 196, 202:221, 225: 2.33, 258

anisotropic, 33p 77biqunzdrxtic, 33claaslcal, 133oonstantz 135, 140, 160, 170, 17% 217,

218, 221, 222, 234y 249, 258direct, 36, 43enera, 16y 4l, 45, 75, 84, 85, 89, 90,

105, 120-123, 125, 133, 134,$37-1391 144, 1V, 157, 159,160, 165, 16:-170, 174-176,179, 180, 207, 216, 217, 2,31,233, 235, 236, 2431 247, 258

clx-kalj 1.34-136, 1381 142, 243indirxt, 36, 48, 77integrzz, 16, 18y 20, 24j 41z 43, 44, 4s,

48, 58, 68, 85> 135, 222lengtk 1431 259, 26l

faanyng model, 202-204Fxvxaay eFeck 60Fe, ece lronFe3 O4, 28, 23257Fe 63, 781FesMaots 31>A,.rml rws, 10

SUBJECT JQDEX 317

Ferrrl1 1eve1, 44j 45ferzic ozdc, sc 7-:Fienoafmvimagnet, 27-30, 32# 44, 99ferrlte. 28, 3l, 43, 44, 86, 91, 25l

bazium? 'ee Balhelaolgcobalt se.e CoFezO4Mn-zu, se.e Mno.6Zn().:$sFoz.:sO*

ferromagnedc resonancey 89, 9l, 135,215-217

feldp 2l6modes, 216, 217, 255

fower state, 260-26% 266Fouder series, 53, 54, 117, 257

p-factor, see Land; fxrrenr'yœecm, 206, 212, 222

Hall elect, HaII probe, 61Hamiltonian, 6, 8, l7, 32, 38, 41, 42, 53,

54, ?0, l09Eeisenberg, 35, 42-45, 48, 59z 60& 62)

70, 75, 1% 79y 82-84, 133Hartree-Foclt, 38, 41Fe, sc coerdvityHcisenbcrg Harnl-ltoniazla >ee Hrnarr,''lto

ni=homogeneous magnetization se.e mag-

netizaiion, uniformhydrog=, 10, 43h e seld, 63, 78hysfzresis. 3, 47, 97-99. 106, 108, 174,

184, 21G 254-25% 262lizrztlng cuz-ve, 1-3, 105, 107, 108,

141, 213, 214, 255-257, 259,261, 263, 266

zninor loop: 2Ycurve, 2

imperfections, 88, 183, 209, 210) 249,265

impurlts 2l, 33, 34, 88: 208initial suscqptibility, se.r susceptibility:

initialiron, 3, % 27, 35, 47,-.4*/, 56, 57. 62, ,J8,

79, 86, E8, 96, 100, 124, 135,141, 143, 169, 171, 172, 205,206, 209, 215, 250, 255

whisker, 58, 209-212, 236Isiag modql, 69-71, 76, 77

onmdimensional, see onœdlmcnsio-nal Isiag model

two-dimenionalz ,ee tw-dimensio-nal Ising modq)

itjnerut electmnsj 3% 44-48, 59

âB, .:e Boltmzrmn's constanKerr efect, *, 61kinedc eergy, 22*Kmnûcker smbol, 37, 49

Lagruge function, 2,2,4Landd factor, 1% 28z 35, 18lLandau

levcls, 10tbeory of phax Mndtions, 63, 64.

68, 69, 75Langevln function, 83, 97, 98Laplace tr=sform, 154, 159Larmor preceslon, 97locxlixqd spin, l0, 36, 44, 46.48, 59, 62,

233low-dimln<ional magnet:m, 76

M, zee marqtszationmaaeiic forcq microscopy, aee. mlcros-

*Wznavetic momeuto ly 5-12. l5y 22, 23.

27, 28, 33, 35, 44*8, 55& 56&83. 84, 92, 93, l06

orbiia), 35magnedc viscodty, 100magnedte, see :F'eaO4magnetization

co don, 137: 141, 147: 148, 155,160: 163: 165, 168, 173, 174,184, 189, 207,-217, 225, 226,229, 230, 238, 2391 242, 244,24% 250, 252, 253, 257-259,261-265

curv'e? 108, 218, 254dircctlon, 84, 85. 87, 89, 90, 92, 94;

97, 99, 100, 105-107, 113,l20z 128, 137, 138, 140, 141,143, 147, 157. 167, 172, 178,181, 189, 196, 215, 218, 219,222, 231, 253, 254, 258

distribution, 47, 112, 1147 123, 124,130, 144, 148-150, 152, 173,2V, 252, 259

procex, 9û, 133, 225, 231, 265r y 99, 185: 188, 191-.193, 202-

204, 206, 207, 209, 225, 231,253

ripple, 89, 108: 259

' 318

sNce dependence, 87, 90, 91s 105,1.08) 165, 169, 174) 185: 190,196) 243

stmcturea 127, 139, 155, 163, 164,166-170, 180, 186, 189, 242,2467 248-251, 256, 260-264

uniform? 95s 1.12-115) 120, 122, 124,125, 728-130, 135, 1.42, 185,188, 2099 214, 225-228, 231-233, 235, 236, 252, 253, 255,260, 263) 264) 266

magnetizations N.:components, 83, 84, 91, 98, 107, 157,

171, l86rem=ent, Mrp 2, 61, 99: 101, 212,

254) 255saturatioa, Mal 2-4: 30v 31, 33p 45I

46, 57-60, 62, 63) 66, 67, 75-78j 101, 106, 112, 120, 123,136, 140, 217, 218, 221, 254,255

vector, 1, 4, 26, 46y 85y 87, 90, 92-94.96, 105) 112) 113, 117, 120,134-136) 1d19, 165) 174, 175,221, 232, 244, 260

magnetocrysvlline anisotropy? 4ee aaî-sotropy, maaetom-ystnlline

magnetostaticenergy', 90, 109: 1t1, 113) 115, 116,

119-129, 133, 7.38-142: 1+1-150, 152, 154-161, 165, 167-170, 7.72, 174, 176, 180, 194,198, 207, 209, 216, 217, 219,226, 231, 233-235, 238-243,246, 247, 250) 251: 255) 258

force? 120, 124, 145, 196interaction, 103, 165, 202) 239, 241potential, 109-112, 116, 118, 119,

122, 126, 128, 130: 131, 148,$50-154, 187: 188: 194, 238,239) 242) 248) 257) 266

problems? 145) 149-151, 173) 176,195, 196) 241) 243: 247: 257,258

magnetostrictiow 87, 88, 100, 141, :75:249

magnon, 55, 58m=ganese, 32, 46) 76, 79Ma:twell's equations? 1, 3: 109, 110, 122,

133, :42, 7.46: 216MBE, 77me= feld, Jcc molecular feld

SDJECT INDEX

MFM, scc microscopy, maaetic forcemicroscopy

Loreatz electrom 169, 171, 257magnetic force, MPM, 61I 252optical, 61, 209nz--annln g e:ectroaj 61, l72traasmislioa elqctron, 203) 262

Mn, s6v maagaaesqMn0.@Zao.3sFea.csO4, 230MrlRi, 2:8, 209Mno, 32molecular beaa epitaxy, ze.e MBEmolecular Eeld. 4, 5, 12, 16-18, 20, 28)

32-34, zlz.j 49, 59, 60, 68, 7%80, 81

Mö>bauer œect, 22, 27, 58j 62j 63, 78,95, 97-100

Mrl Jee maretisation, remanentAGl sa magnetizationv saturation

Néelspikes, 207temperature, TN$ 22-24, 26, 27, 32)

70 I 79theory of aatsferromagtàetîsm, 21, 22theoor of ferrimagnetism, 27) 28theory of surf Me anisotropy, 89

Néel point, se.e Néel temperatttreIneutron depola.rsz-xtionz 230neutroa diAaction, 22. 47, 171Ni, sr.z nicakelnickel, % 42-46, 61, 7% 86) 8% 135) 141,

169: 212: 232: 235NMR, scc nuclear maaetic rœonancenuclear magnetic rœonattce) 22, 63nudeation, 137, 183-189, 191, 194) 196)

200) 201, 205, 207-218, 230,231) 236, 237, 254-256, 259-265

centre, 208, 209feld, 1841 185, 1899 191-194, 196,

198-200, 202-207) 209-211:214) 217: 253-256) 258, 263)265, 267

revm-sal modes Jcc under the mode11Ilnmey

obhte spheroidv 115: 194: 196: 197) 200,204, 232

one-dimenshonalIsing modely 71v 72, 75v 76maaetism, 75I 76

SUBJECT DDEX 319

orbital maretic znoznent. ae,e mareticmoment, orbital

ortNoferrite, 33orthonorrnpl set, 37oxide, 28z 32, 78, 79, 100, 222

parxmmet, pnvxmxgaetîqm 1-4 710-12, 15-17, 22, 24t 27, 78,84, 97-99, l08

partition functjon 55, 71- !Pauu exclumoa prmdple, 37, 40, 43, 46

oyy Vj 89, 1G81 143, 162, 163,166-169, :71-173, 251

pveablt, lj 2, 131, 132, 206.' planar crygtxlllne defects, 2l7

pole avojdauce prindple, 145, l46potential

encgy', 142, 224magnetOtaticj lel mMaet*tatic pp-

tentialprohte spheroidy 115, 116, 193, 194,

200, 201, 204, 205, 231, 232,236, 255, 256

rve o-xr:h, 31, 35, 36, 44, 4a5, 143reiprocity, 148, 149, l76relaxation timej 94..-96, 98: l00j l0l

Browm 94, 95Néeely 94, 96

remaaoce, Jee mognethatioG rema-nent

remormallzation groupj 34h 694 l23Ynxnœ

electron spin, se.e electron sph zoch-nnn çe

fromagnetict ac ferromagneic res-onance

nuclear magnetic, J* auclear mM-nedc nenaace

revewal modew 4* ander mode Aamqsripple stmcture, sa magnetluO-on rip-

ple

nr'nl m' potezttsal) ,c< magnetosvtic po-tentii

s ' bypotbub 69s ' electaron micoscopy, ;co mi-

self-coniutency, 179, 180, 221, 224, 245)248, 25l

self-eergy', zcc mxretostatic energySîO, '?*, 79

speM' c hea.t.y 22j 27, 45, 55z 65, 67: 75,76

spheroida!coordknates, 23lmve functions, 23l

spY, 5, 7. l0, 12, 14-18, 2*22, 24, 25t3l, 33..47, 41-43, 45, 46, 48-53, 56z 63, 69-71) 76, 79, 83-85, 89, 97, 100, 105, 120, 123,133, 134, 136, 137, 143, 145,172, 196, 221, 233-235, 237

devlatioa, 5O, 51, 53operator, 50, 52

waxea, 34, 48, 54, 56, 59, 6O, 78, 109,123, 125

spin-orbit intemcfon, 84, 85, 90, l44spontaueous mMaetiexlrion, aee zzzagne-

tizytion, satutationStonee-Woîtfarth model, 9 ly 105-108,

115, 116, 185, 189. 202, 204,254, 262, 263

supere.x , 43, 44supe tiyrrn, l6, 7% 92-100,

103, 212, 22% 230, 261, 266se=epdbility, 1, '8-11) l5, 22A 91, 108,

132 '

iaitial? 15, 23-27, 29-32, 65-67, 74-.77, 80y 9l, 254

T., se Curie tempœatureT>, &es micrvopy, traasmuion

electron'MC, 76Tw , 4ee N&1, temperaturetransidon metal) 36, 44two-dimensional

1s1R modely 72, 75s 76maretism, 75-77, 79, l64

AAnlfonn mametizatiom Jee magnetiza-tion, Mniform

valencey 3% 45vector potential, 110, 152, 242, 258vortex c ation, 253, 260, 26l

wal: aot domain wallweak ferroma&et-tsmv 331 79wyu'sa 3-5, 17

;'

domdns? 3-5, 85Geld, aee mn:rwnlxlz.r âeld

XY-model, 70

Zeemxn qsqcts 63