医学物理学9Word下载.docx

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9.1BlackbodyradiationandPlanckhypothesis

9.1.1Historicalrecall:

studiesonlight

•Withoutquestion,geometricopticsistheearliesttheorytodescribethephenomenonoflightanditisstillusedtoday.Asyouknow,thereareseveralbasiclawsandequationsingeometricopticsandtheyarestillusefultodayinexplainingtheopticalinstrument.Youmightnoticethatlightpathsareallstraightlinesingeometricoptics.

•Intheearlyof18thcentury,basedontheexistingknowledgeoflightatthattime,Newtonproposedhiscorpusculartheoryoflighttoexplainthelighttravelsalongastraightlineinthespaceoruniformmedium.Insuchatheory,lightisregardedasastreamofparticlesthatsetupdisturbancesinthe‘aether’（whichisrecognizedasthemediumoflightpropagation）ofspace.Newton’scorpuscular（微粒的）theoryisthefirstonetotrytoexplainthenatureoflightbutthereexistmanyotherinterestingandbeautifuleffectsthatcannotbeexplainedbyNewton’scorpusculartheorysuchaslightbendingaroundcorners.Thereforeitisnotacorrecttheory.

•ItisknownthattheGeometricopticsisjustaphenomenon,macroscopicandapplicationstudyonlight.Thatlighttravelsinastraightlineinspaceormediumisjustaphysicalmodelusedtostudylightingeometricopticsanditisthesuperficialphenomenonoflightandithasnothingtodowiththepropertyoflight.

•Thewavenatureoflightisbasedontheexperimentsofinterferenceanddiffractionoflight.SuchaphenomenoncanbeexplainedbyHuygen’sprincipleandthesephenomenaputNewton’scorpusculartheoryoflightintohistory.

•ThecompletewavetheoryoflightwasfinallygivenbyMaxwell（JamesClerk）whoshowedthatlightformedthepartofelectromagneticwaves.ThefamousexperimentstoshowlighthaswavepropertyareThomasYang’sdouble-slitinterferenceexperimentandFraunhoferSingleslitdiffractionexperiment.In1801,ThomasYoungdidadouble-slitinterferenceexperimentoflightandshowedthatawavetheorywasessentialtointerpretthistypeofphenomenon.Andfromthen,Newton’scorpusculartheoryoflighthasbeenconsideredasawrongtheoryfortheexplanationofthenatureoflight.AnotherfamousexperimentwasdonebyFraunhoferSingle-slitdiffraction.Itshouldbealsoexplainedbywavetheoryoflight.

•Inthemiddleof19thcentury,lightsarerecognizedaspartofelectromagneticspectrumanditsspaceandtimedependencefollowsthelawsgiveninMaxwellequationswhichwereconsideredasacompletetheoryforelectromagneticwaves.Therefore,lightbelongstoelectromagneticwavesanditdoeshavethewaveproperties.

•MostphysiciststhoughtPhysicsskyscraperhadbeencompletelybuiltandsomeminorproblemslikeblackbodyradiationandlightspeedwouldbesoonfoundoutusingexistingphysicstheory.However,Geniusphysiciststhoughtthattheblackbodyradiationandlightspeedwerethetwopatchesofcloudsinphysicssky.Infact,the“minor”problemshadshakenthe“skyscraper”ofclassicalphysics.

•Inordertoexplainthequantumnatureoflight,wehavetounderstandhowthefollowingthreeproblemstobesolved:

blackbodyradiation,photoelectriceffectandComptonEffect.Amongthethreeproblems,blackbodyradiationisthemostimportantone.

9.1.2Basicconceptsofblackbodyradiation

Typesofheatenergytransmissionareconduction,convectionandradiation.

Conductionistransferofheatenergybymolecularvibrationsnotbyactualmotionofmaterial.Forexample,ifyouholdoneendofanironrod（铁杆）andtheotherendoftherodisputonaflame,youwillfeelhotsometimelater.Wecansaythattheheatenergyreachesyourhandbyheatconduction.

Convectionistransferofheatbyactualmotionofmaterial.Thehot-airfurnace,thehot-waterheatingsystem,andtheflowofbloodinthebodyareexamples.

RadiationTheheatreachingtheearthfromthesuncannotbetransferredeitherbyconductionorconvectionsincethespacebetweentheearthandthesunhasnomaterialmedium.Theenergyiscarriedbyelectromagneticwavesthatdonotrequireamaterialmediumforpropagation.Thekindofheattransferiscalledthermalradiation.Blackbodyradiationproblemwasfoundintheresearchofthermalradiation.

Blackbodyisdefinedasthebodythatcanabsorballenergiesthatfallonit.Thisislikeablackhole.Nolightsormaterialcangetawayfromitaslongasitistrapped.Alargecavitywithasmallholeonitswallcanbetakenasablackbody（cf.Fig.12-1themodelofblackbodyinyourtextbook）.

Blackbodyradiation:

Anyradiationthatenterstheholeisabsorbedintheinteriorofthecavity,andtheradiationemittedfromtheholeiscalledblackbodyradiation.

9.1.3Twosuccessfullaws

1.StefanandBoltzmann’slaw:

itisfoundthattheradiationenergyisproportionaltothefourthpoweroftheassociatedtemperature.

1.

（9.1）

whereM（T）isradiationenergyandactuallyitistheareaundereachcurve,σiscalledStefan’sconstantdeterminedbyexperimentandTisabsolutetemperature.

2.

Wien’sdisplacementlaw:

thepeakofthecurveshiftstowardslongerwavelengthasthetemperaturefalls.（cf..Fig.12.2thespectrumcurvesofblackbodyradiationinyourtextbook）

（9.2）

isthepeakvalueofthecurveM（T）andbiscalledWein’sconstant.Thislawisquiteusefulwhilemeasuringthetemperatureofablackbodywithaveryhightemperature.Youcanseetheexampleforhowtomeasurethetemperatureonthesurfaceofthesun.

•Theabovelawsdescribestheblackbodyradiationverywell.

9.1.4Problemsexistinblackbodyradiationandthreeformulas

•TheproblemexistingintherelationisbetweentheradiationpowerMλ（T）andthewavelengthλ.

•Blackbodyradiationhasnothingtodowithboththematerialusedintheblackbodyconcavewallandtheshapeoftheconcavewall.

•Twotypicalblackbodyradiationformulas:

oneisgivenbyRayleighandJeansandtheotherisgivenbyWein.

1.RayleighandJeans’formula:

In1890,RayleighandJeansobtainedaformulausingtheclassicalelectromagnetic（Maxwell）theoryandtheclassicalequipartitiontheoremofenergy（能均分定理）inthermotics（热学）.Theformulaisgivenby

（9.3）

where

isradiationpower,C1isaconstantnumbertobedetermined,isthewavelengthofblackbodyradiation,Tistheabsolutetemperature.Rayleigh-Jeansformulawascorrectforverylongwavelengthinthefarinfraredbuthopelesslywronginthevisiblelightandultravioletregion.Maxwell’selectromagnetictheoryandthermodynamicsareknownascorrecttheory.Thefailureinexplainingblackbodyradiationpuzzledphysicists!

ItwasregardedasultravioletCatastrophe（disaster）.

2.Wein’sformula:

Lateronin1896,Weinderiveanotherimportantformulausingthermodynamics.

（9.4）

whereC2andC3areconstantstobedetermined.Unfortunately,thisformulaisonlyvalidintheregionofshortwavelengths.

3.Plank’sempiricalformula:

In1900,afterstudyingtheabovetwoformulascarefully,Planckproposed（提出）anempiricalformula

Fig.9.1Blackbodyradiation

（9.5）

wherecislightvelocity,kisBoltzmen’sconstant,eisthebaseofnaturallogarithms.

Itissurprisingthattheexperienceformulacandescribethecurveofblackbodyradiationexactlyforallwavelengths.

9.1.5DerivationsfromPlanck’sformula

1.RayleighandJeans’formula;

Forverylargewavelength,theRayleigh-Jeans’formulacanbeobtainedfromPlanck’sformula;

2.Wein’sformula;

Forsmallerwavelengthofblackbodyradiation,theWein’sformulacanbeachievedalsofromPlanck’sexperienceformula;

3.StefanandBoltzmann’slaw;

IntegratingPlanck’sformulawithrespecttowavelength,theStefanandBoltzmann’slawcanbeobtainedaswell.

4.Wien’sdisplacementlaw.

Finally,accordingtothebasicmathematicaltheoryanddifferentiatingthePlanck’sformulawithrespecttowavelength,Wien’sdisplacementlawcanalsoderived!

9.1.6Planck’sHypotheses

1.Themoleculesandatomscomposingtheblackbodyconcavecanberegardedasthelinearharmonicoscillatorwithelectricalcharge;

2.Theoscillatorscanonlybeinaspecialenergystate.Alltheseenergiesmustbetheintegermultiplesofasmallestenergy（0=hν）.ThereforetheenergiesoftheoscillatorsareE=nhνwithn=1,2,3,…

3.derivationofPlanck’sfumula（omitted）

4.hνwasnamedphotonbyEinsteinand

（9.6）

iscalledPlanck-Einsteinquantizationlaw.hiscalledPlanckconstant.

（9.7）

Therel