Kramers-Kronig关系

1 Lecture5 Thedielectricresponsefunctions Superpositionprinciple Thecomplexdielectricpermittivity Lossfactor ThecomplexdielectricpermittivityandthecomplexconductivityTheKramers Kronigrelations 2 PHENOMENOLOGICALTHEORYOFLINEARDIELECTRICINTIME DEPENDENTFIELDS Thedielectricresponsefunctions Superpositionprinciple Alineardielectricisadielectricforwhichthesuperpositionprincipleisvalid i e thepolarizationatatimetoduetoanaelectricfieldwithatime dependencethatcanbewrittenasasumE t E t isgivenbythesumofthepolarization sP to andP to duetothefieldsE t andE t separately Mostdielectricsarelinearwhenthefieldstrengthisnottoohigh Thesuperpositionprinciplemakesitpossibletodescribethepolarizationduetoanelectricfieldwitharbitrarytimedependence withthehelpofso calledresponsefunctions LetusconsiderthechangesofelectricfieldfromvalueE1toavalueE2atamomentt 5 1 3 whereSistheunit stepfunction 5 2 Thetime dependentfieldgivenby 5 1 canbeconsideredasthesuperpositionofastaticfield E2 andtime dependentfieldgivenby 5 3 Therefore wefindfromthesuperpositionprinciplethatthepolarizationattimest t duetothefieldgivenbyeqn 5 1 istheequilibriumpolarization E2forthestaticfieldE2andtheresponseofthefieldchangeE1 E2 seefig 5 1 Figure5 1 4 ForalineardielectricthisresponsewillbeproportionaltoE1 E2 sothatthetotalpolarizationisgivenby 5 4 Here t iscalledthestep responsefunctionordecayfunctionofthepolarization Forsimplicityletusrewritethisexpressioninmuchconvenientform 5 5 Att 0 t t in5 4 5 6 Inprinciple bothamonotonouslydecreasingandoscillatingbehaviorof t t arepossible Forhighvaluesoft PwillapproximatetheequilibriumvalueofthepolarizationconnectedwiththestaticfieldE2 Fromthisitfollowsthat 5 7 5 Letusconsiderthecaseofblockfunction Fort1 tt1itequaltozero Thisblockfunctioncanbeconsideredasthesuperpositionoftwofieldswithunit steptimedependence 5 8 Theresultingpolarizationfort t1canbeconsideredasthesuperpositionoftheeffectsofbothunit stepfunctions 5 9 AnarbitrarytimedependenceofEcanbeapproximatedbysplittingitupinanumberofblockfunctionsE Eiforti t t ti Theeffectofoneoftheseblockfunctionsisgivenby 5 9 Sincetheeffectsofallblockfunctionsmayagainbesuperimposed wehave 5 10 Inthelimitincreasingthenumberofblockfunctions 5 10 canbewrittenintheintegralform 6 5 11 where calledpulse responsefunctionofpolarization Theequation 5 11 givesthegeneralexpressionforthepolarizationinthecaseofatime dependentMaxwellfield LetusconsidernowthetimedependenceofthedielectricdisplacementDforatimedependentelectricfieldE 5 12 Forthelineardielectricsthedielectricdisplacementisalinearfunctionoftheelectricfieldstrengthandthepolarization andforthosedielectricswherethesuperpositionprincipleholdsforP itwillalsoholdforD Thus wecanwriteforDanalogouslyto 5 11 5 13 7 with Therelationbetween pand Disthefollowing 5 14 Takingthenegativederivativeof 5 14 wecangettherelationbetween pand D 5 15 Theunitstepfunctionin 5 14 impliesthatthereisaninstantaneousdecreaseofthefunction D t t fromthevalue D 0 1toalimitvaluegivenby 5 16 Incontrastthestep responsefunctionofthepolarizationcannotshow inprinciple suchaninstantaneousdecrease sinceanychangeofthepolarizationisconnectedwiththemotionofanykindofmicroscopicparticles thatcannotbeinfinitelyfast 8 However inthecaseoforientationpolarizationwecanneglectthetimenecessaryfortheintermolecularmotionsbywhichtheinducedpolarizationadaptsitselftothefieldstrength Inthisapproximation theinducedpolarizationisgivenatanytimet 5 17 where isthedielectricconstantofinducedpolarization Wecanrewrite 5 12 inthefollowingway 5 18 Itisthenusefultointroduceresponsefunctions porand pordescribingthebehavioroftheorientationpolarizationforatimedependentfieldandtoconsiderthererelationshipwith Dand Drespectively 5 19 5 20 9 From 5 19 thatnow 5 16 nolongerholds butshouldbechangedby 5 21 Fromcomparisonof 5 19 and 5 20 with 5 14 and 5 15 onecanobtaintheexpressionsfortheresponsefunctionsofthepolarizationinthecasethatthetimenecessaryfortheintramolecularmotionconnectedwiththeinducedpolarizationcanbeneglected 5 22 5 23 Aswasexpected theassumptionthattheinducedpolarizationfollowstheelectricfieldwithoutanydelayleadstotheoccurrenceofaunit stepfunctionintheexpressionforresponsefunctionoforientationpolarization From 5 16 itfollows 5 24 10 Thecomplexdielectricpermittivity LaplaceandFourierTransforms LetusconsiderthetimedependenceofthedielectricdisplacementDforatimedependentElectricfield ApplyingtotheleftandrightpartstheLaplacetransformandtakingintoaccountthetheoremofdeconvolutionwecanobtain 5 25 where 5 26 s i 0andwe llwriteinsteadofsinallLaplacetransformsi 11 Takingintoaccounttherelation 5 20 wecanrewrite 5 26 inthefollowingway Fromanothersidecomplexdielectricpermittivitycanbewritteninthefollowingform 5 27 5 28 Theequation 5 27 justifiestheuseofthesymbol forthedielectricconstantofinducedpolarization sinceforinfinitefrequencytheLaplacetransformvanishes andtheexpressionbecomesequalto Therealpartofcomplexdielectricpermittivity isassociatedwithrealpartofLaplacetransformoforientationpulse responsefunction 5 29 andtheimaginarypartofcomplexdielectricpermittivity isassociatedwiththenegativeimaginarypartoftheLaplacetransformoforientationpulse responsefunction 12 5 30 Letusnowreconsidertherelationshipbetweentimedependentdisplacementandharmonicelectricfield 5 31 We llrewriteinthiscasetherelation 5 25 inthefollowingform 5 32 thatcanbepresentedasfollows where and 5 33 13 Fromtheequation 5 32 itclearlyappearsthatthedielectricdisplacementcanbeconsideredasasuperpositionoftwoharmonicfieldswiththesamefrequency oneinphasewithelectricfieldandanotherwithaphasedifference Theamplitudesofthesefieldsaregivenby Eoand Eo respectively Calculationoftheenergychangesduringonecycleoftheelectricfieldshowsthatthefieldwithaphasedifferencewithrespecttotheelectricfieldgivesrisetoabsorptionofenergy Thetotalamountofworkexertedonthedielectricduringonecyclecanbecalculatedinthefollowingway 5 34 14 SincethefieldsEandDhavethesamevalueattheendofthecycleasatthebeginning thepotentialenergyofthedielectricisalsothesame Therefore thenetamountofworkexertedbythefieldonthedielectriccorrespondswithabsorptionofenergy Sincethedissipatedenergyisproportionalto thisquantityiscalledthelossfactor From 5 34 wefindtheaverageenergydissipationperunitoftime 5 35 where calledalossangle Accordingtothesecondlawofthermodynamics theamountofenergydissipatedpercyclemustbealwayspositiveorzero Itmeansthat 5 36 15 Thecomplexdielectricpermittivityandcomplexconductivity Inaharmonicfieldwithangularfrequency andamplitudeEothedissipationofenergyperunitoftimeinadielectricisgivenby 5 35 Thisequationholdsfordielectricsthatareidealisolators However mostofrealdielectricsshowacertainconductivity leading inafirstapproximation toanelectriccurrentdensityIinphasewiththeelectricfield 5 37 Theelectriccurrentcausesdissipationofenergy AccordingtoJoule slaw theamountofenergydissipatedduringthetimeintervaldtisgivenby 5 38 16 Foraharmonicfield theenergydissipationduringonecycleamountsto 5 39 Hence theaveragedissipationofenergyperunitoftimeduetoconditionis 5 40 Comparing 5 40 with 5 35 weseethatifwedetermine fromtheabsorptionofenergyinadielectricwealwaysobtainthesum 4 sothatwemustcorrectforthecontribution4 duetotheconductivityofthedielectric thereasonforthisistheequivalenceofthecurrentdensityandthetimederivativeofthedielectricdisplacementinMaxwell slaw 5 41 17 Aslongasthecurrentdensityisgivenby 5 38 thereisnoprobleminseparatingtheeffectsofconductionandpolarization since isaconstantthatcanbedeterminedfrommeasurementsinstaticfields However whenittakesacertaintimeforthecurrenttoreachitsequilibriumvalue therelationbetweenthefieldandthecurrentdensityisgivenbyapulse responsefunction I 5 42 Asforthepulse responsefunctionsofthepolarization pandofdielectricdisplacement D thepulse responsefunctionofthecurrentdensity Iisassociatedwithastep responsefunction I 5 43 Analogouslytotherelationbetweendisplacementandtheelectricfield 5 24 afterapplicationtotheleftandrightpartsof 5 42 theLaplacetransformandtakingintoaccountthetheoremofdeconvolutionwecanobtain 18 where 5 44 5 45 Thequantity givesthepartofthecurrentwhichisinphasewiththefieldandwhichthereforeleadstoabsorptionofenergy Hence thisquantityiscomparablewith Thequantity givesthepartofthecurrentwithaphasedifferenceofwithrespecttothefield Thus iscomparablewith ItispossibletocombinethedielectricdisplacementandtheelectriccurrentbydefiningageneralizeddielectricdisplacementD t 5 46 Toensureconvergenceoftheintegral itisnecessarythatE t approachalimitingvaluezerofort fastenough thiscorrespondswiththefactthatthefieldhasbeenswitchedonatsomemomentinthepast TherelationbetweenD t andelectricfieldcanbefoundbysubstituting 5 13 and 5 42 into 5 46 19 5 47 Using 5 43 andthefactthatthecurrentstep responsefunction I 0 1 wecanfind 5 48 where D t thepulse responsefunctionofthegeneralizeddielectricdisplacement isgivenby 5 49 Ifwe llagainapplyLaplacetransformtotheleftandtotherightpartsof 5 47 we llobtain 20 5 50 where 5 51 MakingtheLaplacetransformin 5 51 we llget 5 52 Splittingupintorealandnegativeimaginarypartwearriveat 5 53 5 54 21 TheKramers Kronigrelations TheKramers Kronigrelationsareultimatelyaconsequenceoftheprincipleofcausality thefactthatthedielectricresponsefunctionsatisfiesthecondition 5 55 Itmeansthatthereshouldbenoreactionbeforeaction Letuscon