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