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首页 06 简单电介质与极化率.pdf

06 简单电介质与极化率.pdf

06 简单电介质与极化率.pdf

上传者: zenger2017 2018-03-03 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《06 简单电介质与极化率pdf》,可适用于自然科学领域,主题内容包含SimpleDielectricsandtheSusceptibilityIngeneral,withoutanexternalelectricfi符等。

SimpleDielectricsandtheSusceptibilityIngeneral,withoutanexternalelectricfieldthereisalsonopolarizationConcerningtheoriginofthepolarization,wehavetodistinguishtwokindsofpolarizationOntheonehand,thereisthedeformationpolarizationInthiscase,inthevolumeofmatter(dielectric)consideredthereareoriginallynoelectricdipolemomentsthesearegeneratedonlybythedeformationoftheatoms(ormolecules)byanappliedelectricfieldAnexampleisshowninFigureAneutralatomconsistsofanegativeelectronicshellandapositivelychargednucleusDuetotheappliedexternalfield,thenegativechargesaredisplacedagainstthepositivenucleussothatadipolemomentisinducedInorientationpolarization,moleculedipolemomentsarealreadypresent,buttheyarecompletelydisorderedduetothethermalmotion,sothatnopolarizationoccursInthediscussionofthemultipolemoments,wenotedthatinnuclei,atoms,andmoleculestherearenostaticdipolemoments(oddmultipolemoments)inthelaboratorysystemThereasonforthatistheparityinvarianceoftheelectromagneticinteractionthatis,anychargedistributionhastobepointsymmetric,(rr)Buttherearealsomolecules,eg,ammonia(NH)orwater(HO)withpossessinginternaldipolemomentsThismeansthatina(rotating)coordinatesystemfixedtothemoleculeadipolemaybepresentButthisisnotthecaseinthelaboratorysystem,becauseinthissystemthevariousdipolepositionsareaveragedoutinspaceandtimeduetoquantummechanicalrotationInthewatermolecule(Figure),thehigherchargeoftheoxygennucleusattractsthecommonelectronsandatthesametimerepelsthetwohydrogennuclei(protons)Hence,theoxygennucleusgetsanexcessofnegativechargeAdipolemomentarises,thatis,apolarmoleculeIfonlytwoidenticalatomsformamolecule(eg,thehydrogenmoleculeH),thecommonelectronsstaymostlyinthemiddlebetweenthenucleiMoleculescontainingpolarboundsareinmostcasespolarthemselves(possessingadipolemoment)Ifthepolarities(dipolemoments)oftheindividualbondsinthemoleculecanceleachother,suchamoleculebecomeshomopolar(possessingnodipolemoments)Thus,thelinearmoleculeCOhasnodipolemoment,althoughtheCObondhasastrongdipolemoment(Figure)ButmoleculeshaveadditionaldegreesoffreedominrotationandvibrationBecauseoftherotation,thedipolemomentisaveragedoutinthelaboratorysystem(Figure)Thevibrationmayactsimilarly,forexample,thevibrationofthenitrogen(N)nucleusacrosstheHplaneintheammoniamolecule(Figure)Inthismanner,aparityconservingpatternofthechargedistributionappearsinthelaboratorysystemFrequently,thetimeforarotationcurlislargecomparedtotheinteractiontimeWWHence,onthescaleoftheinteractiontimethepolarmoleculeappearsasaquasistaticdipoleThisentitlesustothe(oftennotquitecorrect)statementthatmoleculeshavestaticdipolemomentsUsinganexternalfield,onecanalignthemoleculardipolemoments(seeFigure)Ifthisalignmentisperformedforliquiddielectrics(liquidwaxorresin),thenthesematerialscanmaintainthepolarizationaftersolidification(electret)TheelectretistheelectrostaticanalogtothepermanentmagnetBut,sincetheneighborhoodtosomeextentisalwaysconducting,anelectretlosesitsoutwardeffectafterafewhoursordaysButtherearealsoferroelectriccrystalsthatexhibitafinitepolarizationwithoutanexternalfieldbelowacertaintemperature(theCuriepoint,similartoferromagnetsinthemagneticcase)ThissituationarisesbecausethedipolesofnearbymoleculesattempttoorientthemselvesparalleltoeachotherduetothedipoledipoleinteractionAparallelorientationismostfavorablefortwodipoles(lowestenergy),aswehaveseeninChapterWithinanensembleofmanyrotatingandvibratingdipolesatasufficientlylowtemperature,thefreerotationsandvibrationsofthemoleculesarechangedcollectivelysothatallofthemarealignedOnetalksaboutaphasetransitionThenew,orientedphaseissimplymorefavorableenergetically(lowerenergy)thanthefreerotationsandvibrationsofthemoleculesAlthoughthebasicinteraction(inourcasetheelectricinteraction)exhibitsthesymmetryofparityandrotationalinvariance,themanybodysystemcanpreferanenergeticallymorefavorablestateinwhichthissymmetriesarenolongerpresent(ie,arespontaneouslybroken)SeeFigureIngeneral,thepermanentpolarizationofaferroelectriccrystaliscompensatedatonceThefreesurfacechargesproducedbythepolarizationattractcompensatingchargesTherefore,ferroelectricitycanbeobservedonlybyadirectedcompression(piezoelectriceffect),inavariationofthetemperature(pyroelectriceffect),orbymeansofarapidlychangingexternalfieldAllferroelectricsarepiezoelectricandpyroelectric,buttheconverseneednotbetrue(eg,thepiezoelectricquartzcrystalsandthepyroelectrictourmalinecrystalsarenotferroelectric)Theconditionfortheappearanceofpiezoelectricityorpyroelectricityistheabsenceofasymmetrycenterinthecrystal:thedirectedcompressionortheheatinghavetocauseadirecteddisplacementofelectronsorionstoleadtoafinitepolarizationButonetalksaboerroelectricityonlyifthispolarizationispresentinthenormalstateandcanberotatedbyanexternalelectricfieldTypicalferroelectricsare,eg,theRochellesalt(NaKCOHOHO),theKHPOsalt,andtheioncrystalBaTiOHence,thepolarizationwillbeafunctionofthefieldintensity:()PPE,where()PBecausetheappliedexternalfieldsareingeneralsmallcomparedtothefieldsactinginsidethecrystalbetweentheindividualions,wemayexpectthatinaseriesexpansionofthepolarizationinpowersoftheappliedfieldonlythefirsttwotermsareofimportanceTherefore,weattemptthefollowingansatzforthecomponentsofthepolarization:,iijjijkjkjjkPaEbEETheindicesrepresentthethreeCartesiancomponentsTheansatzimpliesthat,ingeneral,componentsofthepolarizationinyandzdirectionsalsooccurwhenafieldisappliedinxdirectionThecoefficientsijaandijkbaretensorsoftheranktwoandthree,respectively(withandelements,respectively),andaredeterminedbythespecificpropertiesofthecorrespondingdielectricTheirmagnitudecanbeobtainedfromexperimentItturnsoutthatatnormal(nottoolow)temperaturesandtheusualfieldintensitiesthelinearapproximationissufficientthatis,wemaysettheijkbandallhighercoefficientsequaltozeroTherearematerials(eg,BaTiOandRochellesalt)forwhich()PEisacomplicatedfunctiondependingontheprehistoryofthematerialInFigure,thecomponentsofthepolarizationaredrawnasafunctionoftheappliedfieldThen,thecoefficientsijaaretheslopesofthestraightlinesForijthecoefficientsaregreaterthanforijthatis,thediagonalelementsofthetensoraregreaterthanthenondiagonalelementsAlso,thetensorissymmetric,ijjiaathatis,sixindependentcoefficientsremainforanisotropicdielectrics(eg,quartz)Inparticularlysimplecases,forisotropicdielectrics,ijijaa,thatisiiPaEThenthepolarizationisparalleltotheexternalfieldInthiscasethedielectricsusceptibilityisdefinedbyeaThenePE()ForsuchdielectricsthedielectricsusceptibilityisapurenumberInthiscasethedielectricdisplacementDis()eDEPEEHere,thepermittivityisdefinedbye()Forafixed(constant)permittivitywehavedivdivdivDED()UsingGauss#theorem,wewanttotransformthedifferentialformulationforthefieldintensityandthedielectricdisplacementintoanintegralformWehavedivSVVdAdVdVQDnD()Analogously,theequations(b)and(b)yielddiv()()ppSVVdAdVdVQQEnE()wherethepQaretheinducedpolarizationchargesWiththesetworelations()and(),wecannowinvestigatethebehaviorofthefieldlinescrossinganinterfacewhichcarriesasurfacechargeWeagainsingleoutthediskshapedelementofvolumecontainingtheinterfaceSeeFigureThefrontareasaretakentobesosmallthattheycanbeneglectedThen()nnSdADDAQADnornnDD()ThenormalcomponentofthedielectricdisplacementjumpsbyanamountequaltothesurfacechargedensityCorrespondingtoequation(),surfacechargesstilloccurintheelectricfieldintensity:()nnpEE()WiththeStokestheoremthecontinuityofthetangentialcomponentsfollows,asinvacuum,becauseof(),ie,curlE,()ttCddEsEAEErforallrorttEE()Withtheserelations,wecanconsidertherefractionoftheelectricfieldlinesinatransitionbetweentwodielectrics:Ifthesurfacechargesareabsent,thenttEEandnnDDIntroducingtheanglescorrespondingtoFigure,weobtainsinsinEEandcoscosEEThedivisionofbothequationsyieldsfortheEfieldtantan()andalsofortheDfieldsinsinDDandcoscosDDtantanInlineardielectrics,andDenotingthemediumwiththehigherpermittivityastheelectricallydensermedium,thisrefractionlawimpliesthatinthetransitionintotheelectricallydensermediumtheelectricalfieldlinesarerefractedawayfromtheaxisofincidence(theoppositeofwhathappensinoptics)Example:PotentialdistributionofapointchargeinfrontofadielectrichalfplanewithconstantpermittivityTwodielectricsshareaplanarinterfaceasinFigureIntroducingcylindricalcoordinates(,,)z,theinterfaceshouldbedescribedbyzIndielectric,thereisapointchargeatthedistancezdThepotentialdistribution(,,)zinbothhalfspacesisrequiredThisisapotentialproblemforthewholespacewithDirichletboundaryconditionsatinfinity(()r)ThepotentialiscomposedofpartialsolutionsforthehalfspaceszandzmatchedbycontinuityconditionsatzTheconditionalequationsareDE,zDE,zE,everywhere()()()qzdxyCorrespondingly,forthepotentialwehave(,)z,z(,)z,zIfwefindasolutioninsomeway,thenitistheonlyone,accordingtotheuniquenesstheorem(Chapter)Thechargeliesat(,,)d,andtheproblemisentirelyrotationallysymmetricaboutthezaxistherefore,theangleisnotneededforthesolution:(,,)(,)zzLetustakeapoint(,)PzintherighthandhalfspaceandcalculatethepotentialforitTheeffectofdielectricissimulatedbytheimagechargeqat(,,)dSinceinadielectricwithconstanttherelationDEisvalid,forthepotentialofthetwopointchargesinthehalfspaceweobtain(,)qqzRR()if()Rdzand()RdzarethedistancesofthepointPfromqandq,respectivelyConsideringthechargefreehalfspace,thenthepotentialmaybecalculatedthereundertheassumptionthattheeffectofthedielectricscanbetakenintoaccountbyplacingapointchargeqat(,,)dThen,thepotentialis(,)qzR()Usingtheboundaryconditionsthatresultforthepotentialandthefieldintensityontheinterface,themagnitudeoftheassumedimagechargesiscalculatedThetangentialcomponentoftheelectricfieldintensityandthenormalcomponentofthedielectricdisplacementarecontinuousattheinterface:()()EzEz()()()zzDzDz()Expressingthefieldintensitybythepotential,E,thenforequation()weobtainzzor()()()zzqqqdzdzdzandperformingthedifferentiation:qqq()Asecondconditionisobtainedfromequation():zzSubstitutingand,thenqqq()Theimagechargescanbedeterminedbytheequations()and():qqandqqSo,weobtainthesolutionforthepotentialinthehalfspace(z)(,)()()qzdzdz()andinthehalfspace(z):(,)()qzdz()FigureshowsthefieldlinediagramsforvariousratiosandapositivechargeqTheimagechargeqalwayshasthesamesignasqInthehalfspace,wethusalwaysobtainthepotentialofamoreorlessstrong(positive)pointchargeqTheimagechargeqchangesitssignifInthehalfspacethefieldlinesarerepelledorattracted(bytheimagecharge),dependingontheratioofthepermittivitiesFor,q,qq,weobtainthefieldofapointchargeinthewholespaceFor,q,qq,thefieldlinesinthehalfspacebehaveasiftheyarerepelledbytheimagechargeqFor,q,qq,thefieldlinesareattractedbytheimagechargeqFor,,qqSeeFigureThedielectricbehavesincreasinglylikeaconductorthefieldlinesareperpendicularintheinterfaceandthepotentialvanishesinthesecondhalfspace(compareequation())WewilldiscussbrieflythedensityofthepolarizationchargeinthisproblemAsweknow,itisgivenbydivpPcompareequation(b)SinceintheinteriorofbothdielectricmediaePE,wehavedivdivdivpeeeDPE()()()for,foreqxyzdzz()Atthepositionofthechargeq,thereisadensityofthepolarizationchargewiththevaluepeqdVq()Butwehavetobecareful,becauseattheinterfaceechangesbye()Therefore,attheinterfacethereisstillapolarizationsurfacechargedensityofmagnitude()pPPn()wherezneBecauseiiiiizzzPE()pcanbedeterminedeasilyaccordingto()()()pqdd()Asexpected,thepolarizationsurfacechargedensitytakesthelargestvalueif,anditvanishesif,thatis,ifthereisnointerfaceatallInthecasethelefthandsideisaconductor,and()becomeslim()pqdd()ThisistheresultfortheinducedchargethatweobtainedinExerciseExample:PolarizationofasphereinauniformfieldWeinvestigatethepotentialofanelectricallyneutralsphere,consistingofadielectricmediumwithpermittivity,imbeddedinauniformelectricfieldwiththefieldintensityELetthepermittivityoutsidethespherebeequaltothatis,thesphereisinvacuum(seeFigure)Onthesphericalsurface,polarizationchargesareformedOntheotherhand,theinterioraswellastheexteriorregionofthesphereremainchargefreeThen,thefieldiEintheinteriorofthesphereandaEE,thefieldintheexteriorregionsatisfytheequationsE,E,D,()becauseDE,withconstBecausewehavechosenthecenterofthespheretobetheoriginofthecoordinatesystemandthedirectionofEtobethezaxis,andsincethefieldatlargedistancefromthesphereisuniform,weobtainasr:zEEe,Ez,zEEz()Furtherboundaryconditionsareobtainedforthesphericalsurface:iarrarraDD()thatis,thenormalcomponentofDiscontinuousacrossthesurface,becauseduetodivD,wecanstatethatthesourcesofDarethetruechargesonlyThus,thepolarizationchargesonthesurfacearenotsourcesofD,sothatthenormalcomponentdoesnotmakeajumpFurthermore,forthefieldintensitywehaveiararaEEoriarara()thatis,thetangentialcomponentremainscontinuousAssumingthatthepolarizationisuniform,andparallelandproportionaltotheexternalfieldthroughoutthesphere,wehavegradDEThenforthenormalcomponentofDweobtainnnrrDEeBecauseof(),iarararr()Now,wewanttoinvestigatethealterationofthefieldEarisingduetothepolarizationofthesphereIntheinteriorofthespherethepotentialiscosiiiEzEr()Outside,thesphereactslikeanelectricdipoleinzdirectionwhosemomentiszpThepotentialoriginatingfromthisdipoleisgivenbycoszzzprpzrrrprThus,intheexteriorofthespherethepotentialiscoscoszzaprEzErrrpr()Sincethepartofthepotentialgeneratedbythedipolemomentisproportionaltor,itvanishesforlargerandonlythehomogeneouspartisleftaccordingto()Wesubstitutethepotentials()and()intotheboundaryconditions()and()Eliminatingthedipolemoment,weobtainarelationfortheelectricfieldintheinteriorFrom()weget,withcoszr:cos(cos)coszziiraraprpErErEErrra()Ontheotherhand,from():cos(cos)coszziiraraprpErErEEra()Thus,from()and()thedielectricfieldintheinteriorofthespherecanbecalculated:iiiEEEEE()Thefieldintheinteriorofthesphereisweakenedbythefactor()Now,wedeterminezp,themagnitudeofthedipolemomentFrom()zipEEaSubstitutingthisinto(),wecansolveforzpandobtainzzpEEpaEa()Equation()mayalsobeobtainedstartingfromthepolarization(dipoledensity):eiizzzEEPPEeee,PEOneshouldnotethatthepolarizationP,togetherwiththesusceptibilitye,arealwaysdefinedwithrespecttothemacroscopicfieldinthedielectricInourcase,thefieldinthedielectricisiEThen,weobtainthedipolemomentpbyintegratingoverthevolumeV:zzzVdVPVEaaEpPeeeThedifferencebetweentheinternalandexternalfieldscanberepresentedbythepolarizationiiEEEEEePP()TheweakeningoftheelectricfieldintheinteriorofthesphereduetothepolarizationiscalledthedeelectrificationThemagnitudeofthedeelectrificationisgivenby(),theprefactorofthepolarizationiscalledthedeelectrificationfactorApartfromthecaseofthesphere,thedeelectrificationcanbecalculatedforashallowslabandalongthinrodSeeFigureForaslabofpermittivitypositionedperpendiculartothefieldE,thecontinuityofthenormalcomponentofDyieldsiaDDoriaEEEForthedifference,weobtain()iieiEEEEPThedeelectrificationfactorisequaltoForthethinrodpositionedparalleltothefieldlines,thecontinuityofthetangentialcomponentofEyieldsiEEtherefore,thedeelectrificationfactorisequaltozeroOnecouldwonderthatinthecaseoftheslabweconsideredthenormalcomponentofthedielectricdisplacement,andinthecaseofthethinrodwetookthetangentialcomponentofthefieldintensityOfcourse,thereasonforthisliesinthegeometryDuetothesmallthicknessoftheslabthetangentialcomponentofthefieldintensityisunimportant,apartfromthestrongfieldsattheedgesoftheslabThesameholdsforthenormalcomponentoftheDfieldinthecaseofthethinrodIngeneral,wecanstatethatinauniformfieldanellipsoidwillbepolarizeduniformlyForauniformpolarization,onedefinesiEENPwhereNisthedeelectrificationfactorWitheiEP,ePNIfthesusceptibilityislarge,thepolarizationisdeterminedessentiallybythegeometrydependentdeelectrificationfactorThisexplainsthedifficultyinmeasuringthesusceptibilityforsmallsamples注关键假设()和()可以通过求解满足半径为a的球面以及无穷远处的边界条件(),()和()的拉普拉斯方程得到:i()(cos)lllmmmlllllmlarbrCPe,由于问题是轴对称的,故()(cos)llllllArBrP,利用边界条件,即得球内和球外的解()和()Example:SphericalcavityinadielectricAhomogeneoussphereofradiusaandwithpermittivityisimbeddedinaregionwithpermittivityInabsenceofthesphere,auniformfieldzEEeisinthisregionThepotentialandtheshapeofthefieldlinesinpresenceofthefieldarerequiredForasphericalcavityinthedielectric(,),thedensityofthepolarizationchargeonthesphericalsurfacehastobedeterminedaswellastheelectricfieldproducedbythesurfacechargesSeeFigureThistaskcanbetreatedanalogouslytoExerciseThere,theansatzforthepotentialwasiEz,razpzEzr,ra()Fromtheconditionthatthenormalcomponentsofthedielectricdisplacementarecontinuous,weobtainrararr()Therefore,wehavetoreplaceonlyinequation()ThisreplacementyieldstheconstantsoccurringintheansatzforthepotentialcosiEEErcoscoszpaEEraEr()Thefieldinsidethesphereisalwaysuniform,asperansatzThefieldlinesarerefractedattheinterfaceaccordingtotherelationEfield:tantanDfield:tantanThus,weobtaintheshapeofthefieldlinesshowninFigureTheDlinesarerefractedatthesurface(justliketheElines)TheirnumbercannotincreasebecausedivDsincetherearenoexternalchargesIncontrast,divdivpEPthatis,thenumberoftheElinesmayincreasebecausetheirsourcesarenotonlythetruecharges()butalsothepolarizationchargesdivpPButapolarizationoccursinourexampleThepolarizationdensityistherefore()pPPnPnPnTheinteriorregionofthesphereistheexteriorregionofthedielectricSo,thenormalvectornhastopointoutsidethesphere,thatis,rnrForPitisoppositetherefore,therelativeminussignappearsThepolarizationchargesofthebothpartssuperposeonthesurfaceofthesphereWiththerelationsgradPE,gradPEweobtaincos()praraErrIfthereisacavityinsidethesphere,then(vacuum)Hence,cospEThesepolarizationchargesareindicatedinFigureTheirsignchanges,dependingontheratioThepolarizationchargepgeneratesthefieldintheinteriorofthesphereThepotentialproducedbythesesurfacechargesisthedipolepartof:coscoszDpaErrThispotentialcorrespondstothefieldcosgradsinDDraErrrrEeee(cossin)raEreeNamely,thedipolemomentofthepolarizationchargepisspherecossinzpppzdSaadaEjustasitshouldbeaccordingtoequation()AmolecularmodelofthepolarizabilityInthisandinthenextsections,weconsidertheinterrelationbetweenthemolecularproperties(eg,thepolarizabilityofamolecule)andthemacroscopicallydefinedparametertheelectricsusceptibilityeWediscussthisinterrelationonaclassicalbasisbecausethusourunderst

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