Domain Theory.docx

Domain Theory.docx

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DomainTheory

http:

//www.irm.umn.edu/hg2m/hg2m_d/hg2m_d.html

DomainTheory

Aremarkablepropertyofferrimagneticmaterialsisnotsomuchthattheyhaveaspontaneousmagnetization,butratherthattheirmagnetizationcanbeinfluencedbytheapplicationofverylowmagneticfields.Eventheearth'sfield（50μT）cancausemagnetizationchangeseventhoughtheinteratomicexchangeforcesresponsibleforthespontaneousmagnetizationareequivalenttoafieldofabout1000T,almost100milliontimesgreaterthantheearth'sfield.

Whatallowsthistooccuristhefactthatthesampleisactuallycomposedofsmallregionscalledmagneticdomains,withineachofwhichthelocalmagnetizationissaturatedbutnotnecessarilyparallel.Domainsaresmall（1-100'smicrons）,butmuchlargerthanatomicdistances.

Theexistenceofdomainsishintedatbytheobservationthatsomemagneticproperties,andinparticular,coercivityandremanencevarygreatlywithgrainsize.Thisisbestillustratedinthefigurebelow,whichshowsthevariationofHcwithgrainsize.

Themagneticbehaviorcanbesubdividedonthebasisofgrainsizeintofourranges

1.SPM:

superparamagnetic

2.SD:

singledomain

3.PSD:

pseudo-singledomain

4.MD:

multidomain

ThemaximumcoercivityforagivenmaterialoccurswithinitsSDrange.Forlargergrainsizes,coercivitydecreasesasthegrainsubdividesintodomains.Forsmallergrainsizes,coercivityagaindecreases,butthistimeduetotherandomizingeffectsofthermalenergy.

Domainsconstituteafundamentalconceptinmagnetism.Aferro-orferrimagneticmaterialmaybegenerallydefinedasonethatpossessesaspontaneousmagnetization,Ms,dependentontemperature,butonlyslightlydependentonappliedfield.Thetheoryofferromagnetism,basedonelectronicexchangeforces,predictsthemagnitudeofMs,butsaysnothingaboutthedirectionofMs.Experimentally,itisobservedthatforahomogeneousspecimenatconstanttemperature,themagnitudeofMsisuniformbutthedirectionofMsisingeneralnotuniformfromoneregiontoanother（onascaleofmicronstomillimeters）.Uniformityofdirectionisattainedonlybyapplyingalargeenoughfieldtodrivethedomainsoutofthesample,orbyreducingtheparticle'sdimensionstosmallenoughsizetopreventdomainformation.

Domainsareformedforthefollowingreason.Consideralargesinglecrystal.

Supposeitisuniformlymagnetized,andhenceasingledomain.Surfacechargeswillformontheendsduetothemagnetizationandarethemselvesasecondsourceofamagneticfield（thedemagnetizingfield）.Theenergyassociatedwiththesurfacechargedistributioniscalledthemagnetostaticenergy.Itisjustthevolumeintegralofthefieldoverallspace.

Themagnetostaticenergycanbeapproximatelyhalvedifthemagnetizationsplitsintotwodomainsmagnetizedinoppositedirections.Thisbrings（+）and（-）chargesclosertogether,thusdecreasingthespatialextentofthedemagnetizingfield.

Thissubdivisionintomoreandmoredomainscannotcontinueindefinitelybecausethetransitionregionbetweendomains（calledadomainwall）requiresenergytobeproducedandmaintained.Eventuallyanequilibriumnumberofdomainswillbereachedforagivenparticlesize.

Domainwallsareinterfacesbetweenregionsinwhichthemagnetizationhasdifferentdirections.Withinthewall,themagnetizationmustchangedirectionfromthatinonedomaintothatintheotherdomain.Domainwallshaveafinitewidththatisdeterminedprincipallybyexchangeandmagnetocrystallineenergy.

Let'sconsideradomainwallinwhichthemagnetizationchangesby180?

.Thechangeinmagnetizationwithinthewallcanbegradualasin（a）orabruptasin（b）.

Theexchangeenergyactstokeepspinsparallelandcanbekeptsmallifthe180?

rotationtakesplacegradually,overmanyatomicunits.Therefore,theexchangeenergyissmallin（a）butlargein（b）.

However,thespinswithinthewallarenolongeralignedalonganeasyaxisofmagnetization.Thisproducesananisotropyenergy,whichishighin（a）butlowin（b）.

Theexchangeenergytendstomakethewallaswideaspossiblewhereastheanisotropytendstomakethewallasthinaspossible.Asaresultofthiscompetitionbetweenexchangeandanisotropyenergies,thedomainwallhasafinitewidth（ontheorderof100nm）andsurfaceenergy.

Theinterplaybetweenlongrangeandshortrangeeffectsresultsinthedomainstatesbeinggrain-sizedependent.Inaddition,thenumberofdomainsforagivengrainsizedependsonthemagnitudesoftheexchange,magnetocrystalline,andsaturationmagnetization.Asmentionedbefore,theseconstantsaredependentontemperatureaswellascomposition.Hencedomainstatesindifferentmagneticminerals（magnetiteandhematite）willhaveadifferentgrainsizedependence.Thedomainstateswillalsovarywithtemperatureforasinglegrainsize.However,asaruleofthumb,thelargerthegrainsizethemoredomainsitcontains.

SingleDomain（SD）

Asthegrainsizedecreases,acriticalsizewillbereachedwherethegraincannolongeraccommodateawall.Belowthiscriticalsize,thegraincontainsasingledomain（SD）.AnSDgrainisuniformlymagnetizedtoitssaturationmagnetization.

SDgrainsareveryimportant.TochangethemagnetizationofaMDgrain,allyouneedtodoistranslatethedomainwall,aenergeticallyeasyprocess,whichcanbeaccomplishedinrelativelylowfields.ThusMDgrainsaremagneticallysoftwithlowvaluesofcoercivitiesandremanence.

However,theonlywaytochangethemagnetizationofaSDgrainistorotatethemagnetization,anenergeticallydifficultprocess.Thus,SDgrainsaremagneticallyhardandhavehighcoercivitiesandremanence.HereisanexampleofanSDandMDgrainascharacterizedbyhysteresisloops:

ThecriticalsizeforSDbehaviordependsonseveralfactorsincluding,thesaturationmagnetizationandtheshapeofthegrain.MostestimatesoftheSD-MDtransitionsizearebasedontheoreticalcalculations.Formagnetite,thebestestimateforthetransitionsizeisabout80nm.Herearesometheoreticalresults:

Forhematite,thetransitionsizefromSDtoMDismuchlarger（15μm）,primarilybecausethesaturationmagnetizationisabout200timeslowerthanformagnetite.

Pseudo-SingleDomain（PSD）

ThedistinctionbetweenSDandMDisstraightforward.However,smallMDgrainsexhibitamixtureofSD-like（highremanence）andMD-like（lowcoercivity）behavior.Formagnetite,thisbehavioroccursinthesizerangebetween0.1-20μm.

TherehasbeenmuchtheoreticalandexperimentallyworkonPSDgrains.SomecurrentthinkingisthatsmallMDparticlesthatcontainjustafewdomainsmayactuallyhavedifficultynucleatingdomains.InsomecasesMDgrainsexistinmetastableSDstates.Thetransformationofonedomainstateintoanother,suchasadditionorlossofdomains,iscalltransdomaintransformation.

TheimportanceofPSDbehaviorinmagnetite,isthatthegrainsizerangeforPSDbehaviorcoverstherangeinsizesthatmostcommonlyoccurinnaturalsamples.

Superparamagnetism（SPM）

AsparticlesizecontinuestodecreasewithintheSDrange,anothercriticalthresholdisreached,atwhichremanenceandcoercivitygotozero.Whenthishappens,thegrainbecomessuperparamagnetic.

AnSDparticleofvolumevhasauniformmagnetizationdirectedalongtheeasyaxisofmagnetization.Ifvissmallenough,orthetemperatureishighenough,thermalenergy（kT）willbesufficienttoovercometheanisotropyenergyseparatingthe（+）and（-）magnetizationstatesandcauseaspontaneousreversalofmagnetization.

Forsuperparamagneticparticles,thenetmagneticmomentinzerofieldandatT>0K,willaveragetozero.Inanappliedfield,therewillbeanetstatisticalalignmentofmagneticmoments.Thisisanalogoustoparamagnetism,exceptnowthemagneticmomentisnotthatofasingleatom,buttoanSDparticlecontaining105atoms.Hence,thetermsuperparamagnetism,whichdenotesamuchhighersusceptibilityvaluethanthatforsimpleparamagnetism.

Inresponsetoachangeintheappliedfieldortemperature,anensembleofSPMparticleswillapproachanequilibriumvalueofmagnetizationwithacharacteristicrelaxationtime,firstderivedbyNéel:

where

f0-frequencyfactor（109sec-1）

Ku-anisotropyconstant

v-particlevolume

k-Boltzmannconstant

T-absolutetemperature

TheexponentialnatureoftherelaxationtimeonvandTmakesitpossibletodefineablockingtemperature,TB（atconstantvolume）,orblockingvolumevB,（atconstanttemperature）atwhichthemagnetizationgoesfromanunstablecondition（τ<>t）.Forexample,

t（sec）

21

1

60

1017

Insphericallyshapedmagnetiteatroom-temperature,thischangeinenergycorrespondstoanincreaseinsizefromonly22to33nm!

Innature,fineparticlesofmagnetitecanacquirestableremanenceastheypassthroughtheblockingconditionsastheycoolfromhightemperaturesorgrowauthigenicallyordiagenicallyatlowtemperatures.

InitialSusceptibility

InitialsusceptibilityismeasuredinalowACorDCfield（<1mT）andisdefinedastheratioofM/H.InitialsusceptibilityisduetoreversibledisplacementsofmobiledomainwallsinMDparticlesormomentrotationinSDparticles.Inthelattercase,lowfieldsarenotveryeffectiveinrotatingSDmoments.Therefore,susceptibilitiesinSDandPSDgrainsareusuallylowerthanthatofMDgrains.

However,whatisactuallymeasuredinthelaboratoryistheapparentsusceptibility,χo,nottheintrinsicsusceptibility,χi.Thedifferenceisduetotheeffectsofself-demagnetization.

Remember,insideagrain,theappliedfield,H,ismodifiedb