Solutions of the Steady-state Landau-Ginzburg Equ ation in External Driving Fields

St.John's

Solutions of the Steady-st ate Landau-Ginzburg Equation

in External Driving Fields

@Jenn iferM.Rendell, B. Sc.(Hon.)

Athesis submittedto the School of Graduate Studies in partial fulfillment

of therequirements forthedegree of

MASTER

of

SCIENCE,

Department ofPhysics, MemorialUniversityofNewfound land

December1989

Newfoundland

.+.

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Abstract

Astc ndyst,lll,l!oqnntionIkri\'f~1fromthevnrin tion withres pect to them-dorpa-

nuuctc rAf(i )of11Lundnu-Gina bur g fn-eenergy densityof the form

is considered, where" .,.0, C'2:0,D-::fIIisItsecondrnuk ten sor .This is ageuer- nlization ofpriorworkhyWiult'l'llit;o; dill.(.J.Php l,C214D31-4053(10SS1J, who studied thecnSl~"

=()

aiulC=O. Applied to a nmgnct lc system, itdescribe s the behaviour oftill'tungnctizntion ofHcriticalsystemillthepresenceof an extorun!

tungucticfield "nuducnr11strndllfnlphasetransition.TheLrunluncoc fflcica ts A,Il,lind Can~weaklytcuipc mt ur c dependen t, hilt me consideredcoustuutIleal' thetransition temperutnroT" (the Curiepointiu iungnctic syetc uis ),except for AC((T -1;. ).The grndicnt term nllows fors\lnt i,,1 inhomogeneitiesdue touear- estneighbouriateructlons.Twocasesnrcexamined:C=0(D>0)1I1l11C>0 (D<0)which{'OITI':;pon dto secondandfirst orderphasetrnns itlons,respect ively.

Thesymmet riesofthecquut lon nrc exploitedby thesymmetry rc...lucrionmethod to find exactsolutionsilltermsof variedsyuuuctry variables . Thesesolutionsnrc

inthe form ofkinks, humps,singular ,periodi c,anddoubly periodic solutions.

Thephysicalintcrprctntionoftheseresul tsandoth er calculu tio ns(e.!].t'ller g)' , sus cep tibility]luecxl01\thes e resultsisdiscussed .

iii

Ack nowl edgements

Iwnntto expressIllytlumks10Illy sup-rvisors:Dr..J.A. TUSZy{ISki,now atthe

University(IfAI1 JC'l'la,awlDr.A.M.Grnndluud,nowal. Univcrsitd«IllQuehecit TroisTIi"i l"n's,fol' alltheirlu-lplIudSIlAA1'1lt.ioIlS.Iwnntto givespecia lthanks to Dr.John A.WllitdIl'1111,Dr.Bruce Campbell,DojinngYWllI,unrl~1;H"i t'jSkicrski fortheirtillleiilwulillmhmhlt·t1is l;lIssiCJlIs .IwuuttothankIllyfllluilymulalltuy fri" lLflswhoItil\'~ItI~'ll V(~ITIllld('l's llllH liu J!;aboutlilynhselLl.-llIilldl'lllles sOWl'till!

past ycnr,andIonIlurduuuifurhis llllJlpnrl<luringtill:last mou t hofrevisions.

IwouldIIL~olil«-tothanktheProvinceofNcwfoundlnud

e m

^tilegmduntc srholurs hipthlll111'\]1('(1support methroughIllyMuster's progrmu. Thanks,nlso, to theNuclear nl'sc'a rd lCI'ntl'f,Uuiv•.'rsityofAllcrtn, fortheuscoftheir computer fadlitkos ,

Thisrlocunn-nt wasprepnrodusingI)TE-" This tlll'siscontains iutcrurcdiate resultsthat\\'l'l'f:obtuim-dwiththeaidofhIACSYMA™,~IACSY~IAis11trade- markofSyruholics,Inc.

Contents

1 INTROD UCT IO N

2 SYMM ETRY REDUC T ION 13

3 SOLUTIONSOF THE STEADY·S TATE EQU AT I ON 20

3.1EXile!solut ionsofIF~equation .

3.1.1 COllst,nllt solu t ions. 3.1.2 Ek-tueutnrysolutions .. 3.1.3 Ellipticsolutions. 3.2 Sollltio1l!:ofthew^ficrjlll1linn

3.2.1 Constantsolut ions. 3.2.2 Elumeuturysolutions . 3.2.3 l:..:lipticsolutions.. . 3.2.4 Hyper-el lipticsolu tions .,

4 DISCUS SIO NOFTHE SO LUT IO NS

4.1 ln tcrprc tutlon of the solutions.., 4.2 Cl\lculatiollsbased onthe solutions. .

4.3 Stability .

5 CONCLUS ION S

25 26 26 36 45

47 62 67

68 75

77 77

81

Li st of F igures

Theroh-of1i' 2(I/lilldd('nui\lill~thel'l'gil llls(Ifrenlsolutions. IV IIUlyLeill!t',l!;l'ak(!inthes!lw ]edfI'gion.~.•.

U,4 STSingleuudtriplen-nlroots,D>0,W=0.1

u'.jSTSiup;l,'amitri ple'Tealroots,D<n,w=0.1 Il'.j CD A ('omph'x('(lllj\l~i\t,l~pnirnurl n n'nldoubleroot,.D>0, p=ll.l, r =1.

22 28

29

31

l!" 2SD Twodistinctsingle andHdouble renlmot,D>0,Ii>0,

WI

=

-0.1,11' 1=-0.3 nullU'=0.2 . ... ... . . . ... .. 34

\V~ 2SDTwo distinctsinglenucl adoublen~alrooL,D<0,Ii<0,

WI

=

{)'3,W1=-0.2, nndII'

=

-0.05.

IV"2SD Two rlistiuct»iugloHIllI11doublercul root,D>0,"<0,

1l'1=0.3,^U'1

=

-0.2,endw=-0.05. ..

H'~ 25D Twodistilldsingleam i ndou hle,rculmot,D<0,Ii>0,

IVI=-0.1,11"1=-0.3,and1/!

=

0.2.

WllCQ Complexconj ugatepair andaqundruple_rl~111root,D>0, 35

37

38

II.=±1,'"=0.1 40

10 1V⁶SOTSingle,double,triple real mots, D>0,111=0.1 51 11 WGSOTSingle,double,tri ple real roots,D<0,111=0.1,11=0,-1 52

vi

12 W⁶CDCComplexpai r and a donblccomplex pai r of routs,D>0, po:0.1,fJ=0.2,and11== 0,1

13 11''' C2DComplexpai rnntltwodouble rcnlroots,D>0, r

=

fUn, w=0.2;:; ...

14 W^I12520 Twosingle and two douhlerealroots,D>0, r==0. 113 ,

IV

=

0.3,"'/ll=-1 ..

15 Wr.2S2D Two single nndtwodouhle1"('11]roots,D>0,r

=

0...18, ll'=0.2&,u=(J,l

IG W⁶252DTwo singlemill twodllllhl(~n-nlroot s.D<0,r==0. 113, w== 0.3,,,== -2,-1,0,1.

17 11'''2520 Twosingleaurltwodoublereal roots,D<0, r==0.-1 8 ,

tI'== 0.25.

18 Graphicnl sunuunry oflV·¹sohnious 10 Gn,phirnlstunmnryofW^f•(D>0) sol uti ons 20 GrnplJicalmunmnry oftv⁶(D<0) solutions

List of Tab les

The lldioll or the onepar r nnetcr symmetry~ronp..

SYlIUllct./"rYlll'iahk·sE{3) ..

Synuuotry variablesM(2,1) .

54

57

50

G1

G3

G4

71

rz

73

15 1G 17

Suuuunryofll"SU!lll io llS. Suuuuuryuf 11',]solnlillll~ .

viii

09 70 70

INTROD UCTIO N

Thostudy ofuJa/{Ilt'tic phasetransitions,lindthela-lmv iour ofuiagncti csystems ncur astructuralphasetransi tioni~n topir ofmnj orillt('rt~stineoudcuscdma tt er physics.Lnndau fh'st.Plllllicliout the importance of symmetryillphusctr ansi tions audsuggested that. second orderphase trnusitums conldonlyoccurbetweenphns cs ofdifferentsy m llll' l l' )'

\II.

n1'\1'll1yinvol ving a group-subg roup I.nlllsfo m mli oll.t Phasetrnn sitiousHf1I1l'secondkindluwcIIcoutinuonsdHUi~Cof state across

thetrunsition ,buleachphase is still chnructcrizcdh~'ditfcl'\'ul syuun c t.rjca. In other words ,lL sta ll'where thescaleof corrclnt iousis1\lIlx)lllUh~1is con tinuously ap proache d .lnfk-ld111'.1,1')' thisilltermed nppronchinga7,(~1"\)muss theory. At the phase transition,til<'statesof thetwo phases arethesnme, nuelthere isnolatent hcnt associat ed with thetrallsit illu~

Atnsecondonk-rphnac transitionthe free cncq;.v (potential) and itsderiva tive, t'lltl'Ojl}'arerontiuuons,hutits second derivati ve(e.g.11('" t capi\c il}'),;<;diseontin- nOIlS,ThisistheEhu-nfestclassificat ion,whichapp liestomost phasetra nsitions.

I\Int hcllHlticllll}',then,thephasetransitionpointisasingnl1l\ltyof thefree en- cl'~',F.A rlll1llltil.y,culledthe order par am eter (AI),is definedto describethe

chnngcilltlwstructureofthebod ywhen itpaRsesthrough apha se transition, Ilrouically , thed,,~"k "~ allll' leorn second order"liMe trnusilioll,thecriticalpoin~or lbe gllll-li'111idtrnll.<;jt.ioll,inw,h-,)ljIlOS)'UlIl1l'l rych: ong".

ll' h:v;clrans ilioll.~ofthefirstkind haveIIdiscoufinucustra nsition hchvcclI twophasesor different~)'lI1 l1lO'h~·.,\ttlu-I,hm".!uausition,bodiesin 111'0'!irrercnlsl~lc~nrcillequilibrium, lind there isalale nlIwal,h~·>;t<·rl'~i>;and afiuitechangeinvolumeassociatedwithliretrn n ~il ion .

Theorderparameteris definedsuchth atit iszero'inthe'syuunccricnl'phase (theph asewith a higher symmetry,i.e..more disordered),and non-zerointhe 'uus ym mc tricul'pltaSI!{the phnse with lower symmet ry, i.c. moreordered). The symmctry groupofl'lII:!lphasemust hedifferent,amione is usuallya sub-group of the othe r.The more syunnctrienlphaseuaunlly corres ponds to the highertemper- unuv. Some exampleorder pnrumctcrsnrc:Itconcentration differenceofatomsin alattice ,disJlhll'(:I.~cntfrom nil origina lsitein alattice,amucroseoplcmugnotie momentperunit volume (ferromagnet),nncl amngn cti e momentof the sub-lattice (anti-ferr oml1p;lld).

InIn37Lnudnucombinedsevernltheories:theVandcr\VuHls equati on ofstate (the gas-liquidtrnnsit icn},the Weiss theory[f-rrom ngnots)endthe Curle-weiss theory {nnti-Iernuuugucts] into11.sing lephcnomcuologicul(or meanfield)theory.

Enchofthesetheorieshadassumednil intcmctingsystemCHU}dbe replacedhy a system inallexn-ruulIielel- ifonly thatfield isproperlycllllSl!1I{2J.Anon- inh'ractingsystcrnill1111externalfield cnn beexa ct lysolved;thefit'll!isthen determ ined by11.variationalculculntiou. Itis equivalent1,0selecting, outofall possibleconflgnrnrions,theOIlCthntgivesthelar gest contri b utiontothe pnrtition functio n. In fieldtheol'~',this is calledthe classical or tree upproximution. This means tha tall f1uctnationsarc ignored. So,whilethis type ofthcor~'maybe

'The orderpnmmetcrisuulyzero in thesyuunctticalphasewhen tlu:.rei~110estcrll:L! field.

Thc prcscneeofanextcrnnl fieldwillhediscussedlaler.

qlllllitit lltivc1ywrong (bccnuao ofthefluctuntionsnenr thetransi tion point), itcan be nguide,andis a goodst a r t.ing point formorecomplicatedthoorlos that take these fluctu ati ons intoaccount [21since scaling and sca linglawsnrcstillolcy cd .t Lan dau' sple-nouicnologica ! theory of phase tra ns iti ons 14J is basedonthe assumptiontha t,He ar secon d orderphase transit iontcmpc rnturcs,(t. e. ncara struct ural phase trnnaiti ou orinpartic ular,amagneti cphasetrans ition ],thefree energynellsityIlimy berxpn ndod in a powerseries of theorderpammotor, which is co ntinuousnUll thuscantuke nrhit rru-ilysmal lvalues ncar thetransition point ;

I(H ,T,M)

=

lu(ll,T)+bl( H, T) M+~J'1(H, T)/lJ2+/J-;J( H,T)M~+in(II,T )Ar

where10set stheenergy scale ,(i.e.it is the freeenergydens ity of the disordered phase ).The nrgiuncuts of the potential, external field(H)Ian d temper a t ure(T) can benrbitr urily speci fied, while onlythe par ticularvaluesof theorder parameter (Al )that correspond to ancqulli br'iumstat e(i.e. minimize thc potential]nrc allowed.(N.D.Thisexpans io n docsnottakeaccou nt of thesingula rit y atth e tra ns i tion point;it will he shownlate ritisnot neccssn ry.]Formagneticphase tra ns it ions,whi('h willbetheprimaryapplic at ion in thisth esis,},/repr esen ts the mngnct lzuti ou compone n t alongitgivcndircction {i.e.the projectionofth e magnotieationoloug thcaxis in the directionof thespo ntaneo us magnetization), 11IYPCI5t ~li tlgIdalio,,»donot appl)'heresince thedimensionisnolfom.,\llhaLmarginal cli",cusion,3noth.·crelatiutliuvol viug Ihe critiChlilUl,~xofthe corrclntioulengthexists(31.

tlnothers)"sLellls,Landaucocllic ienbwillII(:rUl'clio ns or thc pml.. urc,aud uoLorlhc cxtclnal field.

Hrepresentsthecxtcrnnlmng nctic field, andT,•is the Curietemperature.

Twoconditionsmusthemet for the potentialtobe a rninumum atparticular valuesofM:

~ = "1

^{+AM +3b:iM1+llAf3}

=

0

o ' f

DM'=A+61IaM+3ll M¹>0

in both the syuuuctricphase(M=0) andthe unaymmctricph ase(AI:f0).The firstimplies the coefficient~lmustbe identicallyzero for allpressures and temper- atnrcs.The second impliesimmediat ely thatinthesymmetricphase,A(H,T)>O.

For the unsymmc tric phnsc, combiningth e two conditionsgivesthelncqunlity:

which is satisfied, regardless of the sign of"3M,ifA(H,T)<O.Thusatthe transitionpointAIr

=

O. Forallabsoluteminimum(notjllSt n relative minimum) to exist , the coefficientB,.{ll ,T)nrust bepositive.Theil the firstcondit ion must be satisfiedforM=0,requiring113t.

=

O.(N.n.thesubllcril' t/1'refers to the transi tionpoint.) IfbAH,T)e 0,then there is n Hue of second orderphase transitionpoints in theHT pla ne. If bat.""0andAt. "" 0 onlyatthe transition poin t, there nrcisolate d (secondorder)tm usit ion pointsilltheHTplane.Only linesofsecondorder phasetransitions arcconsideredhere;lJ:i(H ,T)==O. Now

n

l •is positive, thnsD(H,T)mustbe positiveinth e vicinityofthe transition temperature(11.) and isassumed tohe slowly clmnging,80itis sufficient to

useD(H, Tl , ) .ThecoefficientAcanbewritt enasa(H)(T-T1r(H )),assu ming o(Hl >O.t

Formagneticpha~etransitio ns,it is necessary that un der time reversal[i,e, M... -M)thepotentialbe invariant,so niloddordertermsmusthe ident ically zero.(N.n .Th is isanother reasonfor110lineartermwhe ndiscussingmagnetic phasetm nsirions in the absenceofan external ficld.)

At thecritical point[4J(thepointwher ealine of secondorderphase transitions becomesaline of firstorde rphosetransitions) itis necessary toluwc Acr(H,T)=0 andDer(H,1')

=

0, sincea curveof secondordertransitionsrequir esD>O.Thus forthe state ofthe body tobestable,another order is ad dedtothe expansionof thefreeenergy density ;

whereecr(l ! ,T)>O. Bythesame reasoningCis positivein thevicinity of the critic altemperatureTe"nn ditcan be assumeditisalwayspositive,and approximatelyconstant ncarTer•Itseemsrcnsonnblcth at

n

<0is the line of first orderplmsctmusitions(C>0). At n phase transitionof thefirst kind,f=-fo endOf/OM=0 mustbe satisfiedtogether. Thismean s that111

=

^0,orM'=

-3D/{4C )>0, so

n

<0for aline offirst order phase transitions .Substituting thisback intoeither equat iongivesthe equntion ofthatline,lGAC:::; 3D2. This

IThi~istrue fer1110,;1,but 1I0tnil, transitions.

inturn givesthe new truns itiontempera ture for afirstord er transition,interms ofthe secondorder transitiontcm pcrntu rc:

3D' T

,r

(1 J=1"(11)

+

WaC

Lifshitz and Pitucvskii [4J show thatthecurve of transitionsof the first kind pusscs continuouslyinto the curve ofphase transitions ofthe second kindlit the critical

point(i.e.dT/tl lIiscontinuous},andthatthe twocurvesare discontinuous inthe secondderivativelitthe criticalpoint(i.e.lr1/dH1is discontluuous].

Thediffercnce betweenthe twopolyn otuiulsfo r eachkindof phase trnnsiticn

cnnbe underst oodint,ui t i\'clyhycOllsid erillgthe graph of free energy density vs.themngucriantion. Thefourth order polynomi al, which isquadrati cin },[2,

has11.single well{aboveT,.)that conti nuouslysplitsintondoublewell as the tempera tur e isloweredt.hroughtthctransitionpoint.ThenppCll.1"lI11CCof11pnir of degenerate ground statesiscontinuous . Thesixth onh-rpolynomial,which is cubicinM1,hasasinglewell (an d mayhaveloca lminima)abovethe transition point, and as the tcmporn ture islowered,locnlmini mnform linddrop downIIIthe ground state. The appear ance ofthreedegenera te groundstatesis discon t inuous.

Up tothis poiut, th etreat m entIlli ghocnpur elymeanfield,huttheorder pnramctc rcan lx- considered tobeslowlyvaryinginspa ce;toinclude effccts due to nearestneighbour intcrnction s. Thillillcaisdue to Gi nzlmrg(IU:llCCLandau-

Ginzburgtheory]nnd wnsfirslap p liedtosnpcrecnductcrs(5 I,tForlongwavel ength fluctuatio ns.rollsi(I~Tdcrivn tiresofthelow estorde r.'IhctermsA/V },{andVAl willcont ribnteonl)'tosur f...ccclfcets, andcnubenogjccted,buttermsproport ional to(VAf)2willcontributetobulk volumeeffects:

TIlC!freeenergyis110 11'afunctional:

FIIM); H,T)~^{/ f(M ,VM,H,T)dV. (1.2)}

Inthemostgcucrnleusc,theeouetam ofpl'(J!JOrtionlllil)'Dwillbearealsecond rnnktenso r,whichCllllbetliilgon alizcd. Thethree dingnl1nJelementsD;,(i ::::;

1, 2,3)re presenttileprincipalaxes or Illel:luiCl'.IfD;ispositive,theIntticeis ferromagneticalon gtill.'Ziaxis;ifD,islll~gatiw,the lattice is anti-fcrromag nctie along thcX;exis."I'leu ,nnisolropyclflTtsduetonearestneighbourintcrn.ctions canalsobeineorpomtcd.Itisusef ultosc...leZiwi threspect10thecoefficients Vi, sothatthegtm.lk utter mcanbewritteniliAscnla rtimesthegrOldicntofthe or derpar a meter.

n,

allllO~jth'C )

_{;D[(D,M)'+(D,M)'}

₊

_{(il,M)' 1" D(VM)~}

D;ellncg.ulvc

(1.3)

(1.4)

whereDis a realno n-zero scalarandVi=0D/Vz;'[Here,thex/sarethene w, sca le dllXCS.) The grndieutterm isnowin oneofthetwo aboveforms; the first bein g Euclidean spaceE(3),and thesecond a MinkowskispaceM(2, l )witha pseudo-timevariable .

Theext e rnal fieldrnuyheexplicitlyinclu ded inthefreeenergy densityby add inga linear term(-lliH)[41,whereIiis proportionaltotheexter nalfield H.Thesyrn motry ofthemoresymmetricphn soisthenreduced,sincethe ord er parameteris non-zeroeverywhere.The phosetransitio n pointis no longer discre te atH1rand7;.,but smoolllC'd out.Inparticu lar,the sp ecificheat nolongerhas1\

sharpdisco rtitinnity; it issmearedout. Thefinnl freeenergy densitytoheusedin thisthesis is then:

f(M(i),'VM,H,T )=1.-

"U+ ~.W' + _~BU' + ~CU'

^{+D("MI\. (1.5)}

whe re

A

=

a(H)(T-T,,(")),!. ccH.

Fora secon dordertrnusition

C(H,T,, )

=

0,nut,T,,)>0, lind for nfirstorder transition

C( H,T,,)>0,B(H,T,,)<O.

The Ginzhurgcriteri on{4Jgivesthetempe raturefnnge,Oilbideofwhich the Landau thcory isvnlid.Sothistheoryisusefulclose,hutnottooclose,tothe

transitiontempcrnturc.Insidethis fluctuationrange,thr- fluchu tion sresulting fromtheein gnlar nature ofthe free energyucnrthc phesc trnnsifionnrc domi- nant.N~n.rthetra nsiti nll temperature,largescale correlationsnppcnr- This is manifested,forcxnmple ,bycriticalopalesce nceat the gas -liqnidcriticalpoint whenregionsthe sizeuf micronsf1uctulltecoherently

\2,81,

Inamagnet,diver- genceofthe SlIsc('pti bilityin dicatesthellpp ro i\l::hof thetran ai tiou temperature.

Th econditi o nsforvalidityofthistheo rycanhe moreeasily sat is fied as thecrit ical pointis approached.

There arcothertheoriesthnt modelcrit icalphenomena:e_g,Ising (2 and3 dim e nsions ), Heisenbe rg, andsphcrlcnl models. Ofthese,the2-dimclls jollulIsi n g model is exactlysolvab le,and product's remarkableresultsittphnsc transit io n tem pe ratures.Amcclnurism isneededto compare theresultsofthesevariousth e- ories. Themethod usedistolook Itt howphysical quantities(e.g, hen tcapaci t y, susceptibility ,correla tionlengt h,the correlationfunction, andthemagnetizat ion) chan ge as thetmnsitiou poin tisappro ached.This is writtenas the powerof the redu cedtcmpcrnturc(t=(T -11,)/ 7;,)ncartiletrn ns itlou poiut.Th esepow- ers arccollectivelycal ledthe criticalindices. They arccxpcrlmcuully verifiab le end provide astrnigh tfu rwardwa)' to comparetheresultsof the variousmodels.

llnfortunntcly,hCCHUS CLnnda u-Ginzhurg t!Jl'Ol'Ydoesn't npplyintheHuctuation range, criticalexponentscalculatedusingitnrcnot veryreliable,

The next stepis toflndthe funct io nMth a t minimizesthefreeenergy.Th e

station arypointsofFwith respecttoAlcanbefoundbysolvin gthlsequation:

j

^60M

l

SF= j,IVI- J.+.41I+DM' +CM'-2D !iM =O, OM

o=[)~

- <1. -

D~

which resultsilln nou-Iiuce rpn r-tinldiffcll,utinlcqunlion(POE ):

{

6oM(;}

1

2D ₌-h + AM + DM3 + C},f ~_

OM(i}

(1.0)

(1.7) (1.8 )

(1.0)

Itis im portant toutltethaIitwillbeucc cssnryto cheekwhichfunc tions1\1actually minimi zeF,andthusnrc stablesolutions.

In thc context of qunntumficld theoryIlnrt1!llllud ictl l\.class ofno n-linear field equation s(in),Jink(JW!ildspace)ofthe form:

({)I'{f +JII'H +Olr'1"1 + ~.,,41"+1=0 (1.10)

wherep';0,-1/2,-1 ;1111I1/1E(0,1,2,3).Hefound solitarywnv esolutionsofa classoffieldequmiuns fursyst e mswi thpolynominlsclf-iutcrllcti(Jus whichreduce to plan ewavesohuions whenthe ('oup lingeoustantsl't mlCl ,\are set to zero,He pustula teda planewavevariable11=k~XI'.with!'"k"

=III'

nuddevelope d n generalsolut ion.ForI'=1/2,his solutionis exact ly thefllnnofaeasepresented later(25D;secCf}U:'I.(3.13), (3.14);61D<0).However,hesetthefirstintCUlltio n

10

constant toaer o, aml co nsideredonlyth e casewhen~'I'k!'==m²>0, thus hemissed theother boundedsolutionsdetailed inthe2SDcase,and did not getthesingular solutions,

Khan [10] studiedmagneti c phase transitionsasitbasisfo r understanding criticnlphenom enaHeanalysednonedimensionalversionofoqnntiou(UI)with noext e r nal field,rmd nosixthorderterm(h,C

=

0).Because he workedonth e one-dimcnsionnlcase(i.e.theequationisequivalent to the onlillllrJ' differential cquntirm analysedhere ) ,themethods of intcgrnfion arctin,'same as used ill this thesis. Thus,his solutions rorrc s pond totheelliptic solu tions presented here fo r theC==0cascoAll of his solutionslireperiodic,hiltsomearediscontinuous.

He statesthnt thediscontinuoussolutionsam physically unreal isticlindperhap s thesediscontiu u itblIlayheeliminatedb)'theinclusion o{ higherorder termsin the free energyexpress io n.

IIIamoregeneral contextofthekinetics of a firstorder phase tran s ition with the inclusionofdissipution,Gordon (11Jconsidereda timerlcpcndcut Lnmlau- Ginsbu rgequation (one spntlaldimensio n){orthe cvolntionoftheorderparamete r rpas givenb)':

(1.11) where"=:t -virepresent sa planewnvc lJwttakesthe original POE into a soh..~ able onlhlllrydiffecentin l equation(ODE),risthe Lnudnu-K lml n tnikov damping coefficient,11,11,c:>0arethcLandau coollicicut s, andD isthecoefficient of the

11

inhomogeneityterm.Assum ingkink -li ke bou nd ary condition slim._±oo<p'(s)=0, lim.._oo <p=~hlind lim...._oolP==If'2,where'P1l!Y2Me the minimumfindmaxi- mum,resp ectively, ofthe free energy, hegetsa kink solution.Oneexampleprofile shows theintcrfncc betweentheferroelectricphase1,'1:>0 millnpure-electric phaselP2= 0 wherenis th e propngadon rotc of theinterface.Co mp nrlng his resu lts withcXj1rriulI'lltlll valuesof the Landaucoefficients , he foundsatisfactory agr ee ment betweentheory ruul experiment.

Wintcrni tz dal.[121investi gatedequation (1.9) forsecond orderphasetran- sitions(C

=

0,D>0) in the absence ofan external magn etic field. Thiswork isa threedirucnsioualnnnlogueof t.hecaicu lu rions by Khan(10] disc ussed above, They found a largeclassof symmetryvnrinblcs,lindcorresponding so lut ions. The solu ti ons presentedherewillbesimilarillru nny wnys,hut gcucrn l ly more re- stri ct edsin cethe presenceofthe externalmagneticficlrlloWl~n;tllCsymmct ry of the problem .

Theaimofthis thesisis toanalyselJmoregene ral situat ion wherethe space of theindependent vnrinblcsisthrcc-dlmcneional,andcxtc runl fieldspinynnim- portantrole, hutnotimeis involved.Chapter2will beashort outline onthe symmetryreductionmethodasit appliesto thisequation,nndthe benefits of usingsuchII.metho d , The solution swillhe presented,withsumo discussion,in Chapter 3.A1lI00'Cgeneraldiscussionof thesolut io ns and the cnlcnlntious then possible,is presentedillChapter4.

12

2 SYM ME TRY RED U CTION

This is a briefCorn)"intoS)'ln lllct ryreductionasII.met hod forfindingsolutionsto (sy st ems of)part ialdiffcfcntinJ('(}uatioll!l.Till' equati o nofilltC1"c;;tis:

(2.1)

(N.D.V'hnsbeenused here becausethe pol;nlOllIilll above ispWllOrt,ionnltothe derivativeof thepnlynominlill theIrcc energy dCll!lityequation(1.5). )Itisnil cxnm ple of thenon-IiucurKlcin-Gordoncquurlon, whichhns thegrllern l form:

OM=H(M,(""I) ')

wher etlJl~functlonnlH(.\I. (VM)2) cunbesin M (the !line-Gonion equation), sinAI

+

^sin2M[doub le sine-GordonNllmlKIIl)or,ASinthis case,n polynomialin AI.Symmetryreductionusesthepropc rt il'llflfaLiegroup(Ifcont inuou ssymme- tries ofan equationtIlInnoduccnew, iudq K'lld ('llt symme tryvaria blesand reduce the dimensionalityIII thesystem.Themethodisverygeneraland can be used witha systemofmulti-dimensionalPDE.<;flfJ!~order.It iswelldocumentedby Olver [13).and thischapte rwill only attempt to give:ISmachof nnoverviewof themethodl'L'lis mxxlcrlforthisparticl1blf example.Thecflllat ionis solvedin Eu cl id eansp a ceE(3)nndin Minkowski spnccM(2,I),depending onea ch part ie- ularD[eqns.(1.3),(1A)).In thisexa mple thesymmetryvurinblcsusedreduce thePOE to nn ODE.Thebasic procedu refollows:

13

a) Calclllat.<~andsolvethe syst em ofdeterm iningequatio ns. Thecalculation isalgebraicandtediouswhendoneby hand,butide"lIysuitedtocom p u ter op- eratio ns . Rcccnt ly, sr lllboliccomputerp~ms(f.g. inMACSYMAy'I(U!an d REDUCE(IS])111\\"1,'II(.'CIIdevelopedtoimpleme nttheAlgorithm'! forfiudingthe system ortlehormiu illg('(Illations. HOWCVCT,solvingthe systemordetermining equntion s is quite straight forward.Th e remitisthe Liealgebragwhich is defined bydiffer entialoperatorsgjcalledgenerators.IIIthis case, geucratoreofequation (2.1)in Euclid ean spaceE(3)nrc:

(2.2)

wherei,i,l'E(1,2, 3),nndfiji,istile~Ultiosymmrotl'icLcvi-Civitntenso r.These arcthevectorfieldsortranslationandrotation .Thegenera to rs orequaflon(2.1) in MinkowskiSpliceM(2.1)nrc:

Pj

=

Uj, L3

=

Z2PI-ZIP2,

Po

=

~, K;=-ZOPj-%jPO, (2.3 )

WIKTCj

e

(1, 2).Thenew elementsnrc pseudo-tunetrnnslntjoualllithe Lorentz boost.TheLie elgcbrn ,defincdbyth eLie [commuuuion] brnckct nnd cquatio ue (2.2) and(2.3 ), is:

[P"Po)

=

[Po,Pol

=

(P;, P;)=[1'"Ld

=

0,

[Po,Hi]=-P;,[Pi,Ljl=lijiPki[P;,/(;)=-6i jPo,

(Li,Lj

! =

[Ki,H;)=f,;.L",(Li,A'j )=fij,,1\'1; (2.4 )

14

Tab leI:The ad ionoftheonePArametersymmetrygroup

Gener ators~ "0'

.. " , -

~

Po- t\ {1'O

+

p,1'"Z" 1'3)

PI-~ (1'0,%1 +1',1'20%3)

P,-0, (1'0,%1,1'2+ P, 1'3)

P,=iJ" (1'0,%1,1'2' %3 +p)

L I⁼⁼:r3P, -1"P3 (3"o,%.,Z,eosp-xaSinP,Z'3 00sp+ x , sinp) L,==X,Pa-1'3PI (xo,%,cosP+1'3sinp,x1,x 3cosp- x,sinp) L3 ==:r,PI -1'IP' (.To,1'1cosp-z,sinp,x, eo sp

+

^{XIsinp, 1's)}

](1=-XOP,-xIPO(1'0coshp +Xlsiuhp, ZICOSh p +xos in hp'Z2,X3) 1{,=-.ToP2-x,po (ro eoshp+x2sinhp, XI, X1COsllp +xo sinh p,x3) 1( 3=-z on-.TIPS (.Tocoshp+x3 sillhp,xl11'"xacoshp

+

xosinhp )

wherei,i,1:E (1,2,3) nlld6i jistheKron eckerdelta.N.D.Tilecommuturio nrele- tionimpliestheoperato rsnrcnct in&on i"tnlU"bitTluyfuncti onoftheindepe nden t vnrieblcs.EachgeneratorliEghas nfirstinlegrnl tbntistheone-paramete r3ym·

mctrygroupGjor tl.(' localgroupof pointtransfo rnlo,tio m.th"ttakesone solution intonnewsolutillll.Thenctionofthisgruullis:

wherepisthepurmnctcrofthe group.Thatis,givennsolutioninterms of.1'... IIE (0,1,2,3),anew solutionCMbefoundu:iillgz~ " The nctic noftheone- parame te rgroupforthe11msp a ces arcshowntogetheriuTable1.

b) Fi ndnil possible(dn!lMI)sub-algelarnsnndchoose arepresentati veofeach conjugacyclass.Thi9 iscquivnlcrntofimlul&the optimalsystemofsubalgcbras.

15

Table2: Symmetryvariabl esE(3)

IV"

+

'IV,

=

V'(IV ) IV"

+

'IV,=V'(IV )

Methods for find ingthe subalgcbrasnrc describedby [IG}. The results used in this thesisarc taken Irour [16,17,18).

c) Integ rateeachlinear systemof first orderPDE.~corr esponding tothe suba l- geore,usingthemethod

o r

clmmctcristics[HII. This resultsilla function((x;) (thesymmctry variable} whichis aninvnr junt ofthegroup. The dependentveri- able, (AI),is thenexpressedinterms of thisknownfunction,and anewunknown function(H'):

M(i)=lV(((i )).

Substituting this expression forMin th ePOEwillgcnorutcODE ll of IVin{.

The symmetryvmillblcsused inthis thesis weretakenfromresults Ior nnim plici t, general,and non-linearKlein-Gordonequation. Grundlund ctnl.[201solvedthe problemH(Du ,('\1u)2,u)

=

^{0 for[n}

+

l).di men sioualMinkowski spaceM(n,I), whereHisnunrhit11L1',r function.Theresultsfor this example liregiven inTabl es 2 nml3(WeedWld~andvis annrbitrnryIuucrion].

IG

Tnble3: Symmetry variablesM(2,1) lODE'

("0' _-

11'«+0(11'^{( -}1" (1l'«())

I

r, lV((_V' (lV )

r, 1l'« ~ -V'(Il')

- -

.1'o±.1'1 O= V'(II')

.1'1+(,>(.1'0+%1) W((=-V'(11') (z~

+

zD^I/1 IV((+!I-V(_-1"(1V) (J'~ ^x~)I/1 II'C(+l1V(- V'(Il')

(.T~-x~_zDl/1 11'((+ J1V(

=

l"( W )

The symmetry "ill'inhlesforEnclidcnn spacearctmuslntionnlly,cylindrically and sphericnllyiuvnrinnt,respectively,The tmnslntlonully iuvnriant variablecan bethoughtofliSn plnue wave

t .s

^wherekisthewave vector.

The symmetryverinblcsforMinkowski spacearc501I1(.·\\'hlltmorc interesting.

Thefirst and secondnrctranslutlonally invariant withrespectto pseudo-timeDod space,respectively,The thirdis translationellyinvariant011a plane perpendi cular totheJ'I=TZUline,respectively. The four thdescribesasheet,tileprojcct ion of whichonthe.1'" .1'0

=

Xlpla ne is anarbi traryfunctionofXo

+

XI, Thefifthis cylindrically iuvnrinut.ThesixthdescribesIIhyperboloidsheet,andthcseventh IIhype rboloid.Thefirstthreevar iables cunhr. thoughtofasplanewaves:with.1'0 rep resenting11light-likevector(k'

=

1),XIrcprcscnti ng n space-likevector(P= -1)andXo±.1'1rep resentin g"CClOTS00 thelightcone.While the trnnslntlonally invariant variabk'Smay be anobviouschoice tomake,theothers ,are muchIcs!l

17

obvious. Th is is the powerof the sym metry rc-Iuction method - tocalculate,in a methodicalmnnrer, symmetr}'va riables tlmtapplyto1\11equationor systemof equations.IntheCtlSCSenlcnlntedearlier112J,no externalfield meantthatthere was a muchricherfieldof:i}'lllllletryvariables, An d nf interestingODEs.Thiswas due 10the presenceofscaling symmetries (d ilation!!) which Arc validwhenthe polynomial ucn-liucariryishomogeneous[i,e. forsomefuncti on_P(.\~)="P(¢I».

Thesola tionspresented in the following chapternrc forthe equati ons of the form:

(2.5) corresponding tot:1Cnnnslationallyinvariantanddegeneratesymmetryvariables.

NotethiscqunrionulsolUI!!t"Udiscrete sym metries: {_ -{ nnd {_i{.(N.D.

InthefollowingdUlptl~,the±{equ.(2.5»isabsorbe d in tothe coefficientD.) Wh cnC

=

0,theequationisof Pllin]cl'ctype, hutthegeneralccJlIntionC>0 (and "f;0) is uot.All ODE hasthePaillIen\propertyifits general solution hns nocsscntinlSiUglllllriti{'ll,othrrthan poles,which depen d on the initialconditions /19,21). IfAllthecr itiC/IIpo ints arcfixedfind{ isnotncriticnl poiut ,thecritical point sarcindependent oftheillitinl conditions120J,thesolution is unique,ami completelydctcnniucdbythe iuirialpointsWo=lVau)aml ll'o

=

lVk~{oliD}.

(N.D.ChOOSiJlgn bOllnd11l')' conditionrestrictsthe sym metryof tIlesolut ionto a sm allersymmet ry than thnt ofthe equntienibidf.)ThePninleveproperty isII

necessarybu tuotsufficieJlt conditionforsolutionswithoutmovingessentialsin- 18

gular itics.Allequation with thePainlevc proper tywill most likelybeintegrable in termsof elementary,trnnsccndcntelorPninlcvc transcendentalfunctions. A method oftesting for thePainlcvc property is givcn by[221 nnd aMACSYMA™ program basedon those resultsgiven by 1231.TIIlL~,the equntic n(2.5) willhave analy tic solutionsforC=0, and probably solutions withmovingcsscntinl singu- laritics forC>O.Thus, the following chapterpresents solutions to a less tractable equation,The equationsofthe form:

(2.6)

(correspondingto the cylindrically,sphe ricallyand hyperbolically invariantsym- met ry variables]nsyuip totically have the same solutions as (2.5) ,but also arc not ofPninlcvctype.Itis expectedthesolutions of(2,0) will also have movable algebraic branch pointsIlW\mayormay notluwo analytic solutions.

10

3 SOLUTIONS OF THE STEADY-STATE EQUATION

The equati on thatwill beconcentrated011is

fi' (()=

~(-h+ AW+DW'+CW'),

^{C"O, (3.1)}

whereti'==cPlV/de.Equation(3.1) can immediately heintegratedto

whereSoisa.realintegration constantdefinedbytheboundary surfacee(:l)

=

eo.

The iuitalcondit ionsnrc determinedbythe boundary surface oftheODE:

ThusSois definedas follows:

Theannlyaia of (3.2) is divided into two_ClI.~CS;second orderphasetransitions C

=

0(D>0)and first orderphasetransitionsC>0(D<0).RecallAis positive abovethe transition temperature,and negat ivebelow. Rcul solutions existwhen

li

^r2(e)~O. Theformerhas11complete setof nnnlytic solutions, and thelatt erwhenever n,multipleroot occursinthe polynomial.Forsimplicity,a senledvariable1]

= l>:iiM

is chosen,such that thecoefficientof thc highestpower (withtheexceptionofthe sign) is unity.Thcnewequations tobesolvednrc:

IV",li"(.,)= ,(,\+.IV

+

PIV'

+

II" ),

,=

±1 (3.4) 20

where

and

where

Solut ionsnrc unulyscd011thebasi s of theroot structure of the polynomialon thc righthnnd side ofthe11'"4 (ll'G)equation. Throughout this chapter7/is real, thus the value offischosen to keepfDpositive.Foreach case,D>^{0 denotes} lin(7/)bounded below, andD<0 denotesti,2(7/)bounded above.(N.n.The free energy density equation (1.5) isproportional to equatio n(3.2);

but for physicnlrcusonsdiscussedlater,it is notItproblemth a tsomeofthe grap hs nre unbounde dfrom bclow.} Theshndcd areas shownin Fig.1 arcthe regions that cnnbeintegrat ed to find real solutions.The vcrticnl nxesarc the polynomial functio ntln (I/),andthe horizontalaxis is tim dependent fu nction lV.

For the rcmuiuing graphs,theusualpnlr ofaxes isthegraph(i(7/)YS.the solution lV((i), and the extra(righth nnd)axis gives the grnph 1\12(1})YS.lV«(i).

The two graphs nrc placed togetherto emphasizetheconnecti onbetweenthe real roots ofI,iⁿnudthe limits of thesolutionslV.The roots Wi,r,Pilqi, andother

21

~ . _{" \} J _I , I

\, i

'. ;

\/

, ---; y - - -- -- -

0.0

< :: \

,

\ ,

\

Figure1:Th.·mleOlf,i -: (I,)ill11.·t''flllillill~then'gjullsofn'al!;o h.lilllls.l!"IlII1,r heiutcgratedillthe,;11:\(\1'(1 rrgiuus-

constants(e.g.fl'i,~'i)me understoodtobe local tocnch particularsolution. The arguments(;,andfunctionsR;(lV)havebeen numbered .Each PI;l=:±l.Each (;is understood tulmvc±nssociatcdwith it.This merely gives themirrorimage solut ionandis equivalentto IV--+-lV.This sign is thus dropped. IV is graphed from-1to1 because derivations frommicroscopic Hamiltoniandensitiesrequire this boundfor the series expa nsion[5,12). Each multipleroothas aconst ant solutionassociated withit; thesearc graphed, but notexplicitly includedinthe solutions. Sections 3.1and 3.2detail the exact solutionsend theirparticular graphs.Chap ter4 discusses thesesolutio ns.

Thefollowing termswill he usedto describethc solutionsintho remainderof thischarter.Their definitionsaregivouherefor clarity.There arc two kindsof solitarywaves:bumpsandkinks. Each is alocalised travelling wave.Abump ischara cterizedhrtheSlIlIICasympt oticvalue, lind a kinkhy differentnsymp- totievalues. Theyarc relatedin that a kinkis the derivative ofa blimp.Ithas

1I0tbeen^IlhOWIlwhether thesesolitarywavesareinCaet solitons,which lnw cthe

specialpropertyof retaining their shapeend velocityupon collisionwith other solitary waves[24J.Periodicbounde d solutions willbe called spin wnvca[20Jto emphasize theperiodicchangeinthe magneticlattice of thespin vectororienta - tions.The term 'Iilyr n;' is usedtodescribesolut ions that , except forsingularities, nrc essentially constant between realroots oflV²•When theseconstantvalues nrc differentin various regions, theterm'gap' is used to describethe lnbotwccnregion

23

thatlIla~'beinncecssihlc (corrcepondingtonegativeportions of

lV

²),or may have otherboundedrcal solufione. A nucleation center referstoItlocalizedregionof magne ticorde r,i.e.11bump.

24

3.1 Exact solutions ofH,4equat ion

The solutions ofsecond order phasetmnsiticnsarc foundby unelysiogthe root structureof thefourth·orde rpolynomial:

f('\ +nW+tJll'~ +W~ )

!(W-wd(IV -Il'2)( lV -lI'3)(W-w~) ; (3.G)

"0'=

*'

^('\,o,ft)=

~(.io,-h,~) ^,n

^>0.

The absence of a cubic term (IV^J)gives01\1'condit ionon t,hefonr roots(real or complex):

WI+U'2+ Il'J + W4 = O.

Usingthis to determine11'...thecoefficientscnubewritten

where,\islion-zero to maintainthe generalityof the solutions .Noteaoc II and non-zero,requires each 1"00t{Wi} to bedifferentfromthe negativeof anyother i

'f.

j,Wi1:--Wj' Any complexroots must. occurin conjugatepairs since each coefficient isreal.This elimina tes nnumber of simplercures. Therearcseven caseswhichCUllhe dividcclas follows:onecoustunt , threeelementary and three ellipticsolutions.

25

3.1. 1 Constnntsolu tions

The solutiouscorrespondtothethreeroots or equatiou (3.1)(C

=

0),lindarc:

{

-~( 1

^±

.;'3j)l'~/3 +

^fH(l=FV3i)t';'fJ

W(( )

=

(3.8)

t'~f3_

i1c; l/3

wherethe ccnxtnnt r'eillgiven11)':

andtheintegrat ion constnnt$0can be(Guildhy enbstltut lugtheabo vesolutions into(3.2)withli'o=0,(N.D. lVE:R,but this depends011the nctunlvaluesor theconstnnts.]Wh,m"....0,W({)

=

0,

±i.fAiD.

Thissolu tion also correspo nds to the symmetry varinblc011thelightCOlle~

=

⁷⁰±XI_Themaguctiautionis piecewise coutiuuo usill this case.

3.1.2 Elemcnt n ry solu tions

Each case inthisSl'Ctiuli hasal leastone multiple root.Thesolutionswillbe bum ps,singula r 1111,1periodicsolutions, Therewillhe110kinksolutionsillthelV⁴ case because oftheoxtcrnnl field. Allor theiutcgrnlelIl1t.'tlfor solutionsillthis sectioncanbe(rmllllin

1251.

Singleand triple renl roots(ST)

The rootsarc:

11'1 =101=1113=111,1I1~=-3w;0";tt'E:i, 26

withtheconditio ns"'u=_3w⁴D/ 4<0,A.=_3w²D<0{i.e,belowthe transit ion tempcratur c.]uudthefidelisIi=_2w^JD.Thesolu ti on is (Figs.2,3):

where

lV« d =11'(1-1

~(l ),

^(3.0)

a) WhenD>0 thissolution[cqn. (3.0),Fig.2)is singular nt(~- ~o)=

±tfij

andcouldbe iutcrprctcclnsrepresenti ngamagnetic'tri-lnycr'.Thc mngnct iae tion isconstantatUlex:_h^l/3when(~- ~O)2>D/ (w^2D)lindnbruptlyswitches to -310when(~_ ~u)2<D!(w2n).Increasin gtheflcldwillincrease the difference (4w) inthe relative magneti za tions of the'tri-laycr'struct ure. Maximumand minimumcutoffsnrc Imposed (as n result of finitenessofsp in magnitude )ncar the singuluritlcsliSthe grndientchangesrapidly, Whnthapp ensbetw een these cutoffs isnot revealedthroughthis theory, sincethecontinuu m approximation usedappare ntly hren ks flown inthis regime.

b) whenD<0this solution [cqn.(3.0),Fig.3)isabum pwit h height410end limitwas(~-~u)--+±oo.Theminimum mngnoti za tionis-3w.Itrepresentsa nucleat ioncenter ofmagneti c order separate from theconsta ntmagnetiz ationw throughoutthe rest of thesam ple.Increasingthe field11willincreasetheheight ofth is bump.

2i

-1.0

\

lJ>0,rr=0.1 :~STSinPik· all.ltril'l.~r.~.1roots, Fi~'lrc2:It

28

(, \i" (" )

-10 ~005

0.000

-2.0

,

!

^-0.005

-1.0

-0.010

~o -0.015

-~020

1.0

'-0. 025

2.0

·-O. O~

~o

1 o+_..,...,__ -..--r.ll...,..-....--.--..,....-+-O .0 3S

-1.0 1.0

W(,)

Figllft,:J:Il"~STSingleilllfltripl.·n'a\milt,;.D<0,W

=

^0.1

Acom p lexconjugatepnitanda renldoubletoo t (CD)

The roots are:

It'l = 1/>;1=-P,IDJ=w.i= l~- l+ir) jO1P,rE~,

req uiring_.~o=1,~(1

+

r2)D/4>0,A=,,2(r2 -2)D/2,and thefield11=r2,YJD/2.

Theonlyreal solutionOCCIII1lwhenD>0:

where

(3.10)

Thissolution(Fig.4) is singnlnrnt siuh(:l=-2/r.The Iim"_ :1:"'"H'=p.Sothe magn ctiznricuhJ)is rnlllitnnt overmust ufthe sample,but ncar thesingu1nrity themagnctizfltitlflcbengce exponentiallyallliflips tothe 0PlxlSitc sign.returning

tothecousta ut1l11lgnct izaliun(p).This,OII("ecutoffi'!nreimposed,appearsto beessentially1\bomogmcousfield,with all iubomogcucitynenrthe singu larity.

Increasingthefieldwilldrh'Ctheconstant field1\\\':\)'(romthe zero field.

Two dis t in ctsinglenndndoublerootroot (250)

The rootsore :

"'1,11'2,lI'J=11'.1=-(11'1+w2)/ 2Slllj

30

(, -5.0

lV' (, ) 0.210

0.0

0.00II

-0.250

-0.500

-0. 710

0.0

5.0+- .,--...

-....,..-..,--,-''--,-....,..-.,--,..--+-1.000

-1. 0 1.0

W«( ,)

Figure 4:I{"I CDAl'ulllpl,'x"Illljll~:oll'pair lIlIIlIIn'lI!,luuh1croot,D>^0,

I'=O.l,r=1

31

andA

=

(W]1t'2-3(²)Dt2.Choose11'2<till!andintr oducethe parameter (3.11)

When 6=0 the solutionbecomes (SToqn.(~.!))). Whcnb>0 thepolynomial has a doublerootOiltherightorIdt, (nile isthemirrorimage of the othe r,so assumethedouble rootis011the right ,WI<10). Solutionswith doubleroots011

the left can he obt nincd by rellccting( across zero. WhClifi<0the double rootis between thetwosingle roots ,1II~<^1/1<WI.The analysisofthis section istli\idcd hythe signof0,ISwell as h)' thesign ofD.Thecom mon argumenttf)each is:

(3.12) Thefirstgcucrnlso lution is,forfitD>0:

ll'« , )~

,,+

,/2 (3.13)

III(WI w~)cosh((J+II) 4w'

suchthatwhen

D> (): 1'1

=

1,11'~"'2or IV:511'sing ular

//1

=

-1,WI~lV:510hump D<0: 1/]=1,U'2:5IV:510bump

//1=-1,III~11':5w]bump

andwhere

32

a) '....'11'=1D> 0:mel6>0,the solut ion[cqn.(3.13))isshownin(Fig.5).For

"I=1,thissolutionis singnlMat(3=In (4 ",±v'6)runl,having removedthe singularit ies,representsn mngnclic'bi-layer'.The1im~_*""W(C3 )

=

w.When JII

=

-I,the slJlutioniJllnhump,wit hnnextremum ntC3

=

In {101-U'2)andis nmagneticordernucleation center,This solutionisequivalen ttothe one in(OJ.

Themaximu mht'igiltorthishumpisW-WI.Therenrcuorcnlsolutions between

\('2nndWI·IncrcnsiugtbcmngncticHeldincrousosthe 'SliP'be t ween1172lindWlo and moves thecons tantmngnc tiaationIIpthegraph. The widthofthe portio n betweenthetwo6illgl11nl'iticsis6.(3

=

11l1~ 1.

b) WhenD<0und S<0,thesolution«('(1'1,(3.13)) is shownill(Fig.G).

These nrctwobumpswith cxtremumsatC3=IIl(1lt1- till}nudrespectiveheights WI- w.W-"'21Iilllc,_t"'"U'{C3)=w. ThemagnetizlItion islimited tovalucs between1t'2andWI,nud lmsU11c1entiollcentersofmagucticcrdcratthesnmcpcin t butof dilfercnt umgnltndc undorientntion.Increasingtheexternal field fur ther distorts the1>111111)5Iromthe symmetricpos ifiontlu'Y tukewit hno extern al field, nndmovesthe limit ing\'!llllcs IerthcraWII}'Irourthezero posit io n.

Th e secondgcucrnl sclution, for6/D<0,is:

(3.14 ) such thntwhen

D>0 : 1'2

=

1,

1I' :s

^{11'1orII' ,}:5IVperiodic,singular

33

(,

·12.0

-ao

".0

o.oJ-- - -

1.0

\

\...

~O

-0.005

-0.010

-0.015

-0.030

·~ 035

-0.040

Figure~: \ r-l:!~D'1'\ \11,IislinclSilLV;I,·;'11<1 a duuhll'1'I·:t111"'I.,D>0,~>0,

"'.=-OJ,l"~=-0.:1,1I11[til

=

0.2

"

-12.0

IV'(q)

·O.OOJ

-ac

0.0

0. 002

0.001

0.000

-0.001

-0.002

-0. 003

-0. 004

'. 0 -0. 005

-o. es

0.0

8.0+-- ...,...- .,....-,- -..- .1.,-- ..,...- ...,...- ...--,- -+0.

⁰⁰⁷

-1.0 I.OIV(,)

Fi~,,"~G: 11'12SDTII'Iclj,;lilldsill,(.:l'llllI'Ilcl1tI,~:r";,1I-,. ,l.D <O./j<0,

U'I

=

^0.3,1l'2=-().2,111l/1'"=-0.0 5

D<O: /'2

=

-1,W2SlY:5uriper iodic

e) WhenD>0 and~<0,theso lution{cqn. (3. 14))is sh ownin(Fig.7).This solution isperiodicallysin gularat(3

=

Hl'c~ill ~ ,nudha s rclnrivcminim ums litWI nndnmxiinnrns nttv,.Itre presentsa period ic nrrnngcmcnt,of magn etic dou ble 'Inycrs'once thesingularities havebecuremover],The widt hofthelower 'la yer', bet ween smgularit.ics is1I"- 2nrcsin[4 u,/(wl-

w,ll

lind ofthe uppcr 'lajll.!r' is1I"+2 arc sill[4w/ (W!-W2)]. Eitherofthese goingto zeroisequivalent tofJ-.O. As the field increases,the 'Inycrs'move npm-t,andnwny fro m 1V=O.

d) WhcnD<0undfi>0, thesolution[cqn. (3.14))i~shownill(Fig.S).

Thissolution oscillatesbetwee nthevalues11'1an d11'1withIIperiod of211'" and isnon-singul ar.ItrepresentsaspinWII\"C(cbnnglng between vario usstableand metnstnhlcphnsps).Incrcoeingthcfieldincreasesthe»mplitudcofthes c waves.

3.1.3 Elliptic solutions

Thcrcmainingiutcgmls (themust general cases)nrcelliptic, amithus contain only periodic (andsingulur] solu t ions. Ench iutcgml, all from[26],hasei t herfinupper orloweriutcgmtio» limit;this docs!lut meanIIboundnrycoudiLinn lms beenlost, The solutionsure expressedill termsofthe Jneohinncllipticfuuct.ionslll((,k), cn(,k)andSII(,k).(N.n.tn(,k)

=

SlI(,q/eu{(,k).) TIll)parnruetcr, k,is calledthemodulus ; the co mpiemcn tnry1IIOdlll11Sis1/=~.Generally 0<

3G

(, -12. 0

-8. 0

-1J.002

\

.

./

-'.0

O.OJ - - - - -

'.0

0.0

-0.003

c=

^-0.004

-0.005

-0. 006

Figure'i:11-'12SD Two di:-<t jUflsill~l(';llldal[ulIbh·u-alroot,D>0,f,<0,

lI'l

=

0.3,If':.!=-0.2,nudu-=-fl.OG

:\j

(, -12.0

-ao

,

\

~oco

-0.005

-0.010

-1.0 -~ OIS

-0.020

~o -U 2S

-~030

'.0

^{-0 035}

--0.040

0.0

a. o+-..,....-,...,.--.-..l... --1--...-..,....-..--t.O .0<5

-1. 0 1.0

11'((,)

Fill;ll1"Cs:1\"12SDT\\1.distinetsillJ!;]'·:111,1nd01Jbll~n'a] Will.,D<n,,\>O.

"'I=-0.1.".~=-ll.:1.1111,1u·=1l.2

k<1, if110t,II transformationis employedtomake it so.Her e, thisinequality istruebydefin it ion. Thesefun ctions me one valuedfunctions of (,and are

doublyperiodic,hll\'illgonerea lperiodnm]onecomplex period.Th eperiod s are,respectively,(2A:,4iK.'), (4K:. ,2A:

+

2a.:nl1\HI(4A:,2i K/) .(N. n.Thefunction sn¹(( ,k)has period(2K,2iK').)Theperiodic(u n ction is definedbythecomplete elliptic integral (ofthl)first kind );

(3.15)

a.nd K( k' )=k~' .

ThegrnJllL~[ruve notbccu dr a wnillthissectio n be causeitwouldrequirethree dimensionalgl'aphing, nndbeca usc ofsome techn ical problems. Thetwobasic ellipticfunc tionsSll((, ~')andcn «(,k)lmvc uslimitingfunctions:

511((, 0)=sin(, sn((,l)=tllnh(,

CII(( ,O)=COIl(, ell(, l)=:sech(j and limifingvnlues:

-1SSll«,~")Sl, -l S cn(Ck) :$I,

-oo :5tll( , k) :5 oo.

Thesearenilpci-iodlcfun ctions, sothegraphs urcexpected tohe ei therpe ri odically singular [us ill2SD(D>0,6<0)Fig. 7),hu twithmo dulati ons011thecurve ;

30

orpe riodic andhOlluded(asin2SD(D<0,6>0)Fig. 8),andagainwith modulations011thccurve.Therear c110nulriplc roo tsinthis sectio n,sothere willbeno comtllut solutionsandcobumps. E.~'ICIltinlJy,itisnotnecessaryto secthedctnil aofthese5l'~lltiou.~It.,itis possibletoded uce thegcncrnlshapeand natu reoCeachY"l'h Cromthe workdone ear lier.

Two distinc tcom plexconjugat epairsofroots(2C)

Therootsarc:

(fl+q~_'!1'2 )nI2.These solutionsareilltcnll!~ oCtll« ~ )witlitherealperiod 2,(.

TIleonlyrcnlII(Mutio liis..btnincdwhenD>0:

(3.16)

(3.17) nnd

Itisperiodi cally singular at tll((4,

kd =

-1/&\. Thesesolutions represent u periodic nrrungcmont ofmagneticdoub le'layers'ofdifferently ninguctizcd regions.

These solut ionscuuI"CdUCl~tothe CDcasco Thegraphwillbesimilarto (Fig. 7), butwithou t the 'gnp'betwee n'layers' because thereareItOreal rootstocreate au inaccessible rcgjo.r.

A comp le xconjugat epai r and tw odistinctrealroots (C2S)

The roots nret

11'1>U'z.It'3=

w ;

⁼l'+iq.

P= -~(WI

^-Woz)i

"'1.1"2.]1.1]E!R;WI'11' 2.P,'J:f:.0.

w :

^:f:.W~.

peri od 4C Theurgunmutis:

wit hthese defined constants:

41

(3.1S)

a) WhenD>0,two period icallysingularsolu tions,validforlV( ~ )>WIand

The graphwill he similarto{Pig.7)In cludingthe'gap'of widthWI -W'l_

(3.19)

b) When D<~,twobounded periodicsoluti ons,validfurWI~lV« s)>W'l

These solutio nsoscilla tebe tweenWInudW'l. Therearetwoaolntionsfor one regioni:1this casebecause oftheintog ra uoulimit mcutioncd curlier.Thegra p h willbesimila rill nature10Fig.8. Theywillreducetothesirnpll~rrlHWS(CD) and (250 ).

Fourdistinctrealroots (45)

Theroots nrc:

nnd distinctfromr-ar-h otherami theirncgutivca,WI>0>11'4 'The:Be M isthe general case:

, C(WI

+

"''l)(W'l

+

w:IHu ':1

+

lI'.d (IfI'l

+

11'.\)(11'1

+

11':11

1= G (WI

+

11'1+W3

+",,,) .

42

Thesesolutions arc ellinterms of the elli p ticfunct iollflll'l(G;,k~'I)withrealperiod

(3.21)

with th emodulusP\"t."llbr:

Thearg ument(6can hereal orimogiuru-y, sincesu'(i(,k)

=

-tu ' ( ,k'). All of theseso lutionscnubefound fromllllyothe r,hrthenpproprintechoiceofnrgu mclltj e.g.(3.23) eauI,cgcucrutedbyusing(n

+

K

+

iK.iu(3,22).

a) \VhcnD>0,therenrcIourregions ,t\\'oufwhichproduceeiugular periodic solutions, ilmltwu ufwhiebproduceloundcdperiodic!IOlutions.

Fortherrgiu llsII'.<W( 6)ami11"(6

+

A.:)<UI~,thesolution is:

(3.2'2)

anditissingularlitsn²(a,k3)

=

02"'<1.Thegraph issimilar to Fig.7.

Forthel'cgi()ll.~11',1

s:

W( a)<

ll',

nudW:l<IV(II

+

1\.:)$1l'2,thesolution is:

(3.23)

These solutions me non-singula r and oscill atebetweenW2andW3.Thegrap his similarto Fig.S.

b) Wh enD<0,allthe solutionsarcperiodic, non-singularandoscillatebetwee n their respe ctive bounds.The gra pbsnrcsimilar toFig. 8.

Forth e regio nsW2<lV«r,)~WIandtv2$;H'« r,

+/':" )

<11'11the solutionis:

(3.24)

For ther{'giollsw~<lV«6)~U'3andtIl.1~lV« 6

+/.:.')

<W3,thesolutionis:

(3.25)

All ofthesesolutionsreduce nppm priat,dyto thesolutionsgiven{orsimpler casesin(2SD).

3.2 Solutionsof the11'6equ..tion

Theelcmcu tary :-olntiollsof the cquntion (3.5) :

((..\ +

n1l'

+

,8W¹

+

llV~

+

1V⁰ⁱ)

f(11' - 1V,)(1I' - 11'2)(1V - tllJ)x

l= ± I,I/ =,..-I{;

(IV-",~)(H'- U!5)(IV - tf'G) (3.26)

,,-1

= *,

(..\,0,,8 ,,)

= ~(.~o,- h,~,q) .

whichcorrcspcuds to first01'1\(' 1'phasetransitions, nrc found whenatleas t1Idoub le rootexists. There arcfifteencases,which callbe divklcd into four cntegori cs:

const ant, fiveclcmcntury,fiveellipticami fourhr pcr-elli l' tic solutions.The elliptic and hyper- ellip tic solut ionsfire notincluded here . TheLnndnn coefficient D, infill butone cnsc(CDC),isshownto he explicitlynegative ill the elementa ry solutio ns.All sollltimrlfireexp['C5SC(1ns

<

=«IV),sinceitisnolonger possible to iuvcrtthem.Th ismenusitisneces s ary tohe\'('1")'carefulaboutnny±signs associatedwithmdicnlsfindperiodicity [branches]of Iuuctinns illthe implicit solutio n.Thelmsicsu lutiollisI-Iandwill give only onepor tionof the solution. Properconsiderat ion ofthe±sign amiII::'willgivethecou iplctosoluti on. This is pnrficulnrlyilllpor tull1 forbumps nnd!H'l'ieJ(!ic solut ions. Ttrc roots canbe rea l or complex, hut1111)'complex roots mustoccurillct>llj ug ntc pairs ,sincethe coefficients lirerenl.NonUwffls.'imnptiollsli remnde couccr uing theroots.Since

45

there:isnol)uiUl icterm(IV! )therootsllIustl;atis£;r:

WI

+

"'2+11'3+1I1~ +1I'5+"'6=O.

There:isulso uo cubic term(lV³).thustwo ruotJicnu hewritten:

where

-(1111+ 1"1)lI'i-(11'1+",,)'U'3+Uf~^-11'1",, (11',

+

1/11)+'l~l /("'1+'''' +II'3+UI~),

field).Usingthis condition,theeoefllrieuts r-un hewritten:

,\ <= lI'III,~".ll1'~{(/Ill

+

II',

+

11'3+fI1~)1_

Q 'J .

_(11'1+lr'2) ( " '1

+

"'3)('1'1

+

U'~){ "'2+ "'J)(^1rl2+"'~)(II'3

+

"'~)

i::

0,

(WI+1Il1+lr3+'I'~ 1

tJ

~ [(II'~ + Ir~

⁺

II'~+

"'_ 4:1)(11'1 +"'1 +"'3+

Il'~l

+( II'~

+

/f'~

+

",~+1I,~)1_2(II'i'+",~+II'~

+"':)

(3.2; )

(' CI , CI) )]

+ "'; ,-

^"2)(II'llj"

+

^{U'l lt'l}

+

^!l'lll':,)+(w1-

'2

^(!I'IU'l

+

U'~ !I1:1+WJU'1 ,

..., =

lI'11I'2+II'J(/I"+"'2)+lr~(II'I+"'2+"':l1

with). generallylion-zero,Note,again,nC(hbeinglion-zerogenerallyrequires eachroot {Ill;}tohedifferent fromthe negativeofnny.uhc ri

#-

j,lVi

#-

-wi' The field cnu heexp ressedillterms ofthe roots:

11=C(Wl

+

WZ)(WI

+

W;I){WI

+

w~)(Wt

+

ll'3)(Wt

+

tt'4)(ll'3

+

1ll.1)

i'

0, (3.28) G(lI'l+Wt+1l'3+1l'1)

3.2.1 Co ns t antsolutio ns

Thefiveconstant solutionsarc givenhytherootsof (3.1)lindthe integration constant$0cnn then he foundby substitutingthosesolu tions in(3.2). Natur nlly, the roots must nll he real,sinceItrealsolutionis sought.These represent homo- genoousfields. Again,thesymmetry variableon thelightCOllC,givespiece-wise continuous solutions.

3.2.2 Elementa r y solutions

Acom plex conjuga te pairanda quadr up lerealro ot (CQ)

The rootsarc:

"'I='11"1

=

11'3

=

m,1=W, "'s

=

w;'=w(2

+

i);0

i'

11'E!R. Thc coefficients nssocintcd withW⁶require$0

=

5Cwⁿ/G,A.=5CIV~>0(abo ve thetrnnslt ion temperature],while

n

=-lOC w¹/ 3<0uud the fieldish

=

-sewS/3. TIll!IllJlyrealsolutionoccurswln-nD>0, (Fig. 9).The urguruent is:

snrh thn t

-11'$ 11":JI=-1 11'$-11': /1=1.

Thi~solutio n is singulnr at11'=-II'.Trnncuringthesiul;lIll1titics[eaves1\homo- gcncous mngncrlc fidd,prolMwtiounlto theexter nulfl,'hl.

Single,doubleandtr iplerealroots (SD T)

Theroot slin':

11'1

=

~II', "'1

=

11'3

=

-411', "'~

=

lI'~

=

11'11

=

1/1;0'#^II'E R.

Thecoefficientsor1I.t,require"0

=

40Clr^fi/3 ,.-l

=

5~C,v5>0[nbovc the trausl- tion temperatu re},whileD=-20CII"<0mul thefieldis"=3GClI'5>0, since

a) Wht'llD>O.thesolution,[Fig.lll),is:

(1\

=

/'t=lu lll,'1W (1i'1I'-711'+/13...15V<tV-5U1)( 11'-1Il)) 1

3vO,I +4w

48

(3.30)

(3.31)

-0. 0002

0.ססOO

-0.11II01

-0.11II03 IV'(.) 0.11II01

0+- -..- ....---.- ...-.l...;- ...---...- .,...,...-+-0.1lOO4

1.0 II'(C,) C,

-lS.0

-211. 0

-15.0

-1~0

-5.0

e.e

5. 0

10.0

15.0

211.0

25.

-1.0 0.0

Fip-lln'9:11"';C(l('''IlII'~'xr,,"juga'"Il;,irall'1 a((lm,lrul'l..n-ul""'I.D >O. II=±l."' =O .l

where cnch regionisgiven hy:

WS-4w: l/l=1 -4w5W5lP: l/l=1 5w511': I'l

=

^-1,

and11=±Igives the two halves ofeachsolnt.inll. There lire three areas of solutions;thetwooutsidewhichnrosingularnnd themiddle region-4w5IV.5III

whichhas kinks.Theheight ofthe kinks incrcusca nsthcfieldincreases.

b) WhellD<0,theouly real solution(F ig. 11)is:

(II=

3~llrCsill (~ - 2{11~~"4U'))

1I1t 1 5ll' -II'

+

3../5

+ '2

TV-/II (3.32)

whereIIis uu integer .Thetwo halW'Jiof thcsolutionnrc(lffn=0)nnd-(M{n= -1),the solidandclottcd Iiues,respec!h'I'lyill Fig.11.Itis validillthe region IV.511'.5 5wnuddl'SCrilJl'l:Itlnunp withlU'i~ht4l1'. Th isI"I'pr CSI'lltS;}lIudcntion sitc ofmugucricordor.

Acomple x pairandadoublecom plexpni rorroots(CDC) The roots arc:

50

-0. 004 -ll.ool

~002

(, IV'(, )

-10 0.002

0.001 -2.0

0.000

"\

~ool

-\.0

,

- :

:

-0.005

2.0

l.o+--.,...---.,.-~-_.,....:..._-_

... -.,...-+o.ooa

-1.0 1.0

11'(,)

Figut\.·ItJ:11";SOTSill~~"c1"1I1J\."Iripl.-n-nlf\lllb,D>O. II'

=

^11.1

~1