Correlation of energy and free energy for the ther

arXiv:quant-ph/0210209v131Oct2002CorrelationofenergyandfreeenergyforthethermalCasimirforcebetweenrealmetalsV.B.Bezerra,G.L.KlimchitskayaandV.M.MostepanenkoDepartamentodeF´ısica,UniversidadeFederaldaPara´ıba,C.P.5008,CEP58059-970,Jo˜aoPessoa,Pb-BrazilTheenergyoffluctuatingelectromagneticfieldisinvestigatedforthethermalCasimirforceactingbetweenparallelplatesmadeofrealmetal.Itisprovedthatfornondissipativemediawithtemperatureindependentdielectricpermittivitytheenergyatnonzerotemperaturecomprisesofthe(renormalized)energiesofthezero-pointandthermalphotons.Inthismannerphotonscanbeconsideredascollectiveelementaryexcitationsofthematterofplatesandelectromagneticfield.Ifthedielectricpermittivitydependsontemperaturetheenergycontainsadditionaltermsproportionaltothederivativesofεwithrespecttotemperature,andthequasiparticleinterpretationofthefluctuatingfieldisnotpossible.Thecorrelationbetweenenergyandfreeenergyisconsidered.PreviouscalculationsoftheCasimirenergyintheframeworkoftheLifshitzformulaatzerotemperatureandopticaltabulateddatasupplementedbytheDrudemodelatroomtemperatureareanalysed.Itisdemonstratedthatthisquantityisnotagoodapproximationeitherforthefreeenergyortheenergy.Aphysicalinterpretationofthishybridquantityissuggested.Thecontradictoryresultsintherecentliteratureonwhetherthezero-frequencytermoftheLifshitzformulafortheperpendicularpolarizedmodeshasanyeffectivecontributiontothephysicalquantitiesarediscussed.Fourmainapproachestotheresolutionofthisproblemarespecified.Thepreciseexpressionsforentropyofthefluctuatingfieldbetweenplatesmadeofrealmetalareobtained,whichhelpstodecidebetweenthedifferentapproaches.TheconclusionisthattheLifshitzformulasupplementedbytheplasmamodelandthesurfaceimpedanceapproacharebestsuitedtodescribethethermalCasimirforcebetweenrealmetals.PACSnumbers:12.20.Dc,42.50.Lc,65.50.+mI.INTRODUCTIONTheCasimireffectisararemacroscopicmanifestationofthezero-pointelectromagneticenergy.Itresultsfromthealterationofthezero-pointspectrumbythematerialboundaries(seetheoriginalCasimir’spaper[1]andex-tensivereviews[2,3,4,5,6]).InrecentyearsconsiderableattentionwaspaidtotheprecisionmeasurementsoftheCasimirforcebetweenmetallicsurfaces[7,8,9,10,11,12,13,14,15].TheresultsofthesemeasurementswereusedinRefs.[16,17,18,19,20]toconstrainthehypotheticalinteractionspredictedbymanyextensionstotheStandardModelandalsoinnanotechnology[21].ThiscalledfornewtheoreticalinvestigationoftheCasimirforcewithallowancemadefortherealisticboundaryproperties,i.e.,surfaceroughness,finiteconductivityofametalandnonzerotemperature(seeRef.[6]forthereview).Alsothecombinedeffectofthesefactorshasattractedconsiderableattention.InthispaperweconsiderthecorrelationbetweenenergyandfreeenergyfortheCasimirforceactingbetweentwoplaneparallelplatesmadeofrealmetaloffiniteconductivitykeptatnonzerotemperature.AtzerotemperaturetheinfluenceofthefiniteconductivityoftheboundarymetalontotheCasimirforcewasexaminedinRefs.[22,23,24,25]onthebasisoftheLifshitztheoryusingtheopticaltabulateddatasupplementedbytheDrudemodel.InthisapproachtheCasimirenergywasrepresentedbyanintegralwithrespecttocontinuousfrequency,asitisatzerotemperature.However,theopticaltabulateddataandthevalueoftherelaxationparameteroftheDrudemodelatT=300Kweresubstituted,whichareactuallytemperaturedependent.Bythismeans,infact,somehybridquantitywascomputed,whichisdifferentfrombothenergyandfreeenergy,withnoclearrelevancetoeitherofthem.WhenitisconsideredthattheCasimirenergybetweentwoplatesisgenerallyusedtocalculatetheCasimirforceintheexperimentalconfigurationofasphere(sphericallens)aboveaplate,theresolutionofthisissueisofgreatinterest.Belowisshowntherelationshipbetweenenergy,freeenergyandaquantitycomputedinRefs.[22,23,24,25].InvestigationofthecorrelationbetweenenergyandfreeenergyforthethermalCasimirforcehasalsoassumedgreatimportanceinconnectionwiththecontradictoryresultsobtainedbydifferentauthorswhenapplyingtheLifshitztheorytorealmetals[26,27,28,29,30,31,32,33,34,35].Thecontradictionsariseonwhetherornotthezero-frequencytermoftheLifshitzformulafortheperpendicularpolarizedmodesofelectromagneticfieldcontributetophysicalquantities.Therearefourmainapproachestotheresolutionofthisissueintherecentliterature.a)AccordingtoRefs.[26,35]basedontheimmediateapplicationoftheunmodifiedLifshitzformula,thezero-frequencytermofthisformulafortheperpendicularpolarizedmodesisequaltozerointhecaseofrealmetalsdescribedbytheDrudemodel(remindthereaderthatforidealmetalsthereflectioncoefficientsforbothpolarizationsareequaltounityatzerofrequency,andhencethezero-frequencytermforbothmodesisnotequaltozero).ThisapproachleadstotheconclusionthatthermalcorrectionstotheCasimirforcearelarge,negative,andlinearin1temperatureatsmallseparations,andtheasymptoticCasimirforcebetweenrealmetalsatlargeseparationsistwotimessmallerthanforthecaseofanidealmetal(withnoregardfortheparticularvalueoftheconductivity)[32].b)FromthestandpointofRefs.[28,32],theperpendicularpolarizedmodesgiveanonzerocontributionatzerofrequency.Tofindit,aspecialmodificationofthezero-frequencytermoftheLifshitzformulawaspro