Kramers-Kronig关系

1Lecture51.Thedielectricresponsefunctions.Superpositionprinciple.2.Thecomplexdielectricpermittivity.Lossfactor.3.Thecomplexdielectricpermittivityandthecomplexconductivity4.TheKramers-Kronigrelations2PHENOMENOLOGICALTHEORYOFLINEARDIELECTRICINTIME-DEPENDENTFIELDSThedielectricresponsefunctions.Superpositionprinciple.Alineardielectricisadielectricforwhichthesuperpositionprincipleisvalid,i.e.thepolarizationatatimetoduetoanaelectricfieldwithatime-dependencethatcanbewrittenasasumE(t)+E’(t),isgivenbythesumofthepolarization’sP(to)andP’(to)duetothefieldsE(t)andE’(t)separately.Mostdielectricsarelinearwhenthefieldstrengthisnottoohigh.Thesuperpositionprinciplemakesitpossibletodescribethepolarizationduetoanelectricfieldwitharbitrarytimedependence,withthehelpofso-calledresponsefunctions.LetusconsiderthechangesofelectricfieldfromvalueE1toavalueE2atamomentt‘:)t'S(t(t)121)(EEEE(5.1)t'tP1P2E2E1EP0030100tS(t)=tS(t)whereSistheunit-stepfunction:(5.2)Thetime-dependentfieldgivenby(5.1)canbeconsideredasthesuperpositionofastaticfield,E2,andtime-dependentfieldgivenby:t'tP1P2E2E1EP00)(1)E-(E(t)E21ttS(5.3)Therefore,wefindfromthesuperpositionprinciplethatthepolarizationattimestt’duetothefieldgivenbyeqn.(5.1)istheequilibriumpolarizationE2forthestaticfieldE2andtheresponseofthefieldchangeE1-E2(seefig.5.1).Figure5.14ForalineardielectricthisresponsewillbeproportionaltoE1-E2,sothatthetotalpolarizationisgivenby:tt,ttEEEP(t)212(5.4)Here(t)iscalledthestep-responsefunctionordecayfunctionofthepolarization.Forsimplicityletusrewritethisexpressioninmuchconvenientform:)t()(E)t(P(0)P(t)0(5.5)Att=0(t=t’in5.4)10)((5.6)Inprinciple,bothamonotonouslydecreasingandoscillatingbehaviorof(t-t’)arepossible.Forhighvaluesoft,PwillapproximatetheequilibriumvalueofthepolarizationconnectedwiththestaticfieldE2.Fromthisitfollowsthat0)((5.7)5Letusconsiderthecaseofblockfunction.Fort1-ttt1thefieldstrengthisequaltoE1,andfortt1-tandtt1itequaltozero.Thisblockfunctioncanbeconsideredasthesuperpositionoftwofieldswithunit-steptimedependence:1111)(ttSEtttSEtE(5.8)Theresultingpolarizationfortt1canbeconsideredasthesuperpositionoftheeffectsofbothunit-stepfunctions:)tt()ttt(EP(t)111(5.9)AnarbitrarytimedependenceofEcanbeapproximatedbysplittingitupinanumberofblockfunctionsE=Eiforti-ttti.Theeffectofoneoftheseblockfunctionsisgivenby(5.9).Sincetheeffectsofallblockfunctionsmayagainbesuperimposed,wehave:iiitttttti)()()(EP(5.10)Inthelimitincreasingthenumberofblockfunctions(5.10)canbewrittenintheintegralform:6dt'ttt==dt'tt'ttttt)'()'()()'()(EEP(5.11)where,calledpulse-responsefunctionofpolarization.Theequation(5.11)givesthegeneralexpressionforthepolarizationinthecaseofatime-dependentMaxwellfield.)'()'(ttttppLetusconsidernowthetimedependenceofthedielectricdisplacementDforatimedependentelectricfieldE.)(4(t)(t)tPED(5.12)Forthelineardielectricsthedielectricdisplacementisalinearfunctionoftheelectricfieldstrengthandthepolarization,andforthosedielectricswherethesuperpositionprincipleholdsforP,itwillalsoholdforD.Thus,wecanwriteforDanalogouslyto(5.11):DE()(')(')tttDttdt'(5.13)7(')(')DDttttwithTherelationbetweenpandDisthefollowing:DpttStttt(')(')(')114(5.14)Takingthenegativederivativeof(5.14)wecangettherelationbetweenpandD:Dptttttt(')(')(')4(5.15)Theunitstepfunctionin(5.14)impliesthatthereisaninstantaneousdecreaseofthefunctionD(t-t’),fromthevalueD(0)=1toalimitvaluegivenby:ttDtt'lim(')()1(5.16)Incontrastthestep-responsefunctionofthepolarizationcannotshow,inprinciple,suchaninstantaneousdecrease,sinceanychangeofthepolarizationisconnectedwiththemotionofanykindofmicroscopicparticles,thatcannotbeinfinitelyfast.8However,inthecaseoforientationpolarizationwecanneglectthetimenecessaryfortheintermolecularmotionsbywhichtheinducedpolarizationadaptsitselftothefieldstrength.Inthisapproximation,theinducedpolarizationisgivenatanytimet:4/)1(t)((t)inEP(5.17)whereisthedielectricconstantofinducedpolarization.Wecanrewrite(5.12)inthefollowingway:)(4)()(tttorPED(5.18)ItisthenusefultointroduceresponsefunctionsporandpordescribingthebehavioroftheorientationpolarizationforatimedependentfieldandtoconsiderthererelationshipwithDandDrespectively.DsssporttStttt(')(')(')1(5.19)Dsssportttttt(')(')(')(5.20)9From(5.19)thatnow(5.16)nolongerholds,butshouldbechangedby:ssD'tt)'tt(lim(5.21)Fromcomparisonof(5.19)and(5.20)with(5.14)and(5.15)onecanobtaintheexpressionsfortheresponsefunctionsofthepolarizationinthecasethatthetimenecessaryfortheintramolecularmotionconnectedwiththeinducedpolarizationcanbeneglected.psssporttStttt(')(')(')1111(5.22)psssportttttt(')(')(')111(5.23)Aswasexpected,theassumptionthattheinducedpolarizationfollowstheelectricfieldwithoutanydelayleadstotheoccurrenceofaunit-stepfunctionintheexpressionforresponsefunctionoforientationpolarization.From(5.16)itfollows:ttp