ScottHughes 10May2005 [610952]

ScottHughes 10May2005
Massa chusetts Institute ofTechnology
Department ofPhysics
8.022 Spring 2005
Lecture 23:
Magnetic materials
23.1Magnetic ¯eldsandstu®
Atthisstageof8.022,wehaveessentiallycoveredallofthematerialthatistypicallytaught
inthiscourse.Congratulations |younowhaveasolidgroundinginallthemajorconcepts
ofelectricityandmagnetism!
Thereremainsonemajor,importantsubjectwhichwehavenotdiscussedindepth:the
interactionofmagnetic¯eldsandmaterials.Atthispoint,wedon'tevenhaveareallygood
understanding ofabarmagnetworks|anunsatisfyingstateofa®airs,giventhatthisis
thewayinwhichwenormallyencountermagnetismintherealworld!
Onereasonwehaveavoidedcoveringthissubjectisthatitisnotreallypossibletodiscuss
itproperlywithingettingintoadetaileddiscussionofthequantummechanicaldescription
ofmatter.Thewayinwhichmatterrespondstomagnetic¯eldsistotallydeterminedbythe
quantummechanicalnatureoftheirmolecularstructure,particularlytheirelectrons.
Nonetheless,wecanmakesigni¯cantheadwayinunderstanding theinteractionofmag-
netic¯eldsandmaterialsbycombiningwhatwehavelearnedsofarwithasomewhatap-
proximate,qualitativedescriptionofhowmaterialsrespondtomagnetic¯elds.Themain
conceptswewillneedaresummarizedinthefollowingtwosubsections:
23.1.1 Electron orbitals
Theelectronsinamoleculeexistinorbits.Veryroughly,wecanpictureanorbitasasimple
loopofcurrent:
Aloopofcurrentlikethisofcoursetendstogenerateitsownmagnetic¯eld.Inmost
materials,thereareanenormousnumberoforbitslikethis,randomlyorientedsothatthey
producenonet¯eld.Supposeanexternalmagnetic¯eldisappliedtosomesubstancethat
containsmanyorbitalslikethis.ThenettendencycanbeunderstoodintermsofLenz's
law:theorbits\rearrange"themselvesinordertoopposethechangeinmagnetic°ux.This
tendencyforLenz'slawtoworkonthemicroscopicscaleendsupopposingthemagnetic ¯eld
fromthematerial.
23.1.2 Intrinsic magnetic momen toftheelectron
Oneotherquantummechanicalpropertyofelectronsplaysanextremelyimportantrolein
thisdiscussion:electronshaveabuilt-in,intrinsicmagneticmoment.Roughlyspeaking,
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thismeansthateachelectronallonitsownactsasasourceofmagnetic¯eld,producinga
dipole-type¯eldverysimilartothatofcurrentloop.
Becausethis¯eldisassociatedwiththeelectronitself,itdoesnotexhibittheLenz'slaw
typebehaviorofthe¯eldthatweseefromtheorbits.Instead,themostimportantbehavior
inthiscontextisthefactthatamagneticmoment~mplacedinanexternal¯eld~Bfeelsa
torque:
~N=~m£~B:
Inthiscase,theactionofthistorquetendstolineuptheelectrons'magneticmomentswith
theexternal¯eld.Inthiscase,you¯ndthatthemagnetic ¯eldisaugmentedwithinthe
material.
23.1.3 Whichwins?
Tosummarizetheabovediscussion,thequantummechanicalnatureofelectronsinmolecules
leadstotwobehaviors:
²Lenz'slawonthescaleofelectronorbitalsopposesmagnetic¯eldsfromenteringa
material.
²Magnetictorqueactingontheindividualelectronsaugmentsmagnetic¯eldsinama-
terial.
Thesetwobehaviorsareincompleteoppositiontooneanother!However,bothoccur,and
bothareimportant.Anaturalquestiontoaskis:Whichofthesetwobehaviorsismore
important?
Theanswertothisquestionvariesfrommaterialtomaterial,dependinguponitsdetailed
electronicorbitalstructure.The¯rstproperty|Lenz'slawontheorbitalscale|plays
someroleinallmaterials.Inmanycases,thisistheendofthestory.Suchmaterials
arecalleddiamagnetic.Adiamagneticmaterialisonewhosemagnetization (tobede¯ned
preciselyinamoment)opposesanexternalmagnetic¯eld.Eventhoughalmostallmaterials
arediamagnetic,thee®ectissopunythatitistypicallyverydi±culttosee.Aconsequence
ofthisisthatadiagmagneticsubstancewillbeexpelledfromamagnetic¯eld.
Forsomematerials,thesecondproperty|alignmentofelectronmagneticmoments
|winsout.Suchmaterialstypicallyhaveseveralelectronorbitsthatcontainunpaired
electrons;theorbitthushasanetmagneticmoment.(Inmostdiamagneticmaterials,the
orbitsare¯lledwithpairedelectrons,sothattheorbithasnonetmoment1.)Suchmaterials
arecalledparamagnetic.Aparamagnetic materialhasamagnetization thataugmentsan
external¯eld.Paramagneticsubstancesarepulledintoamagnetic¯eld.Inthevastmajority
ofcases,paramagnetice®ectsarealsosopunythattheycanbarelybeseen2.
Oneparticularclassofmaterialsactsessentiallyasparamagneticmaterialsdo.However,
theyrespondsostronglythattheytrulybelonginaclassoftheirown.Thesearethe
ferromagneticmaterials.
1Bearinmindthatthisisahighlyqualitativediscussion!
2Therearesomeinterestingexceptions. Oxygen,O2,turnsouttobeparamagnetic. However,the
randommotionofgaseousoxygenwipesoutanyobservablee®ect.Ifwecoolittoaliquidstate,though,
theparamagnetism iseasilyseen.
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23.2Somede¯nitions
Acoupleofde¯nitionsareneededtodescribewhatwearegoingtodiscuss.Foremostisthe
magnetization ~M:themagneticdipolemomentperunitvolumeofamaterial.Noticethat
magnetization hasthesamedimensionsasmagnetic¯eld:
~M=magneticmoment
volume
=current£area=velocity
length3
=current=velocity
length:
Amaterialwithconstantmagnetization ~MandvolumeVactsjustlikeamagneticdipole
with~m=~MV.
Thesecondimportantde¯nitionisabitofawierdone:itisanewkindofmagnetic
¯elddenoted~H.This~Hisde¯nedintermsofthe\normal"magnetic¯eld~Bandthe
magnetization ~Mvia
~B=~H+4¼~M:
Inotherwords,whenthereisnomagnetization, the\normal"magnetic¯eld~Bisjustthis
new¯eld~H.Thismeansthat~Hisnothingmorethanthebitofthemagnetic¯eldthat
arisesfrom\normal"electriccurrents,asopposedtothosethatare\bound"inmatter.
23.2.1 More details on~H
Wecandosomemorestu®with~H,thoughthisisreallybeyondthescopeofwhatweneed
forourdiscussion.Ifwetakethecurlof~H,weget
~r£~H=4¼
c~Jfree
whereJfreeisthedensityoffreeelectricalcurrent.Thecurlofthemagnetization de¯nes
boundcurrents:
~r£~M=~Jbound=c:
Puttingthesetogether,wehave
~r£~B=~r£~H+4¼~r£~M
=4¼
c~Jfree+4¼
c~Jbound
=4¼
c~J:
Inotherwords,thetotalcurrentdensityisgivenbyaddingthefreecurrentdensity(the
currentduetomovingchargesthatwecanimaginecontrolling)andtheboundcurrentdensity
(thecurrentthatis\boundup"inelectronorbits,andisreallyapartofthematerial).
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23.3Magnetic susceptibilit y
Awidevarietyofmaterialsexhibitlinearmagnetization,meaningthatthemagnetization ~M
dependslinearlyonthe¯eld~H:
~M=Âm~H:
ThequantityÂmisthemagnetic susceptibility.Fromthis,wehave
~B=~H(1+4¼Âm):
IfÂm<0,themagnetic¯eldisdecreasedbythematerial.Thiscorrespondstodiamag-
netism.IfÂm>0,themagnetic¯eldisincreasedbythematerial.Thiscorrespondsto
paramagnetism.
23.4Ferromagnetism
Roughlyspeaking,ferromagnetism isparamagnetism onsteroids.Itskeyfeatureisthatit
isnonlinear:the~Mand~Hdonothaveasimplelinearrelationbetweenoneanother.This
leadstosomesurprisingresults.Inparticular,itmeansthatoncethe\external"¯eld~H
isturnedo®,themagnetization ~Mremains!Thisofcourseishowpermanentmagnetsare
made.
Thereasonsforthisnonlinearbehavioraresurprisinglycomplicated,andaredi±cultto
explaincorrectlyinasimplemanner.(Itisironicthatpermanentmagnets,probablythe
mostfamiliarmanifestationofmagnetism,turnouttobesodi±culttoexplain!)Itisn'tso
hardtodescribethemostimportantfacetsofferromagnetism qualitatively;thehardpartis
explaining whythosequalitativebehaviorsemerge.That,sadly,isfarbeyondthescopeof
8.022.
23.4.1 Ferromagnetic domains
Sinceferromagnetism isconceptuallyalotlikeparamagnetism, weexpectthatitwillbe
explainableintermsofthealignmentofmanyatomicscalemagneticmoments.Thisis
indeedthecase.However,withferromagnetism thereisanimportantmodi¯cation:one
¯ndsthatthemomentsofmanyatomsormoleculestendtobealignedinsmallregions.
In\normal"paramagneticmaterials,themomentsarerandomlyarrangeduntilanexternal
¯eldcomesalongandalignsthem.
Ifyouweretolookatthemagneticmomentlayoutofatypicalferromagnetic materials,
youwould¯ndsomethinglikethis:
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Enoughatomicscalemomentsarealignedthatde¯niteregionsordomainsofmagnetization
exist.Eachdomainisfairlysmall,andeachisrandomlyorientedwithrespecttoitsneighbor;
thematerialasawholehasnonetmagnetization. Thisiswhyarandompieceofiron(a
representativeferromagnetic material)isusuallynotmagnetized.
Ifthismaterialisplacedintoastrongenoughexternalmagnetic¯eld,thedomainsare
re-alignedsothattheyallpointinthesamedirection:
Oncetheexternalmagnetic¯eldisremoved,itturnsouttobeenergeticallyfavorablefor
thesedomainstoremainpointinginthisdirection.Thedomainsare¯xedwiththismagne-
tization.Thisishowpermanentmagnetsaremade.
23.4.2 Hysteresis
Thenonlinearrelationshipbetween~Hand~Bmeansthatthemagnetic¯eld~Binaferro-
magneticmaterialhasarathercomplicateddependenceonthe\external"applied¯eld~H.
As~Hisvariedbetweenaminimumandamaximum,one¯ndsthat~Binthematerialtraces
outaninterestingshapecalledahysteresiscurve:
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NoticethatwhenH=0,B6=0|re°ectingthefactthatthematerialhasbeengivena
permanentmagnetization. Youmaywonderlookingatthisplotwhichofthetwopossible
valuesitwilltake|thepositiveBorthenegativeone?Theanswerdependsonthehistory
ofthemagnetization. IfHwaslargeandpositivemostrecently,Bwillgotothepositive
valuewhenHissettozero.IfHwaslargeandnegative,Bwillgotothenegativevalue.
Itturnsout(thoughit'sabitbeyondthescopeofthiscoursetoshowthis)thatittakes
workto\goaround"thehysteresiscurve.Aswegofrom+Hto¡H,wehaveto°iplots
ofmagneticdomainsaround|ittakesenergytodothat!Theamountofworkthatis
doneingoingfrom+H!¡H!+Hisproportionaltotheareaofthehysteresiscurve.
Oneconsequenceofthisisthatsolenoidswithanironcorecanbeverylossyelementsin
ACcircuits.AnACcircuitiscontinuallychangingthedirectionof°owofthecurrent,and
henceofthe~H¯eld.Ifaportionofthecircuitcontainsaferromagnet,someofthepower
putintothecircuitwillbelostbycontinuallychangingtheorientationoftheferromagnet's
domains.
23.4.3 Curie temperature
Nodiscussionofferromagnetism iscompletewithoutmentioningtheCurietemperature.The
Curietemperature,TC,isatemperatureabovewhichamaterialverysuddenlyceasestoact
likeaferromagnetic. IfthetemperatureisaboveTC,therandommotionofthemagnetic
momentsissostrongthattheycanneverbecomealigned.Justatinybitbelow,though,
andtheycanalign!It'ssomewhatamazinghowstrongthetransitionisatthistemperature.
Foriron,TC=770±Celsius.
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