Blackbody Radiation. All bodies at a temperature T emit and absorb thermal electromagnetic radiation. How is blackbody radiation absorbed and emitted?

Transcription

1 All bodis at a tmpratur T mit ad absorb thrmal lctromagtic radiatio Blackbody radiatio I thrmal quilibrium, th powr mittd quals th powr absorbd How is blackbody radiatio absorbd ad mittd? 1

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3 A blackbody is a prfct absorbr of radiatio A simpl blackbody is giv by a hol i a wall of som closur Both absorptio ad missio ca occur Th radiatio proprtis of th cavity ar idpdt of th closur matrial 3

4 Th y axis is rgy / tim / ara / wavlgth 4

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6 Wi s displacmt law λmaxt Wavlgth dcrass as T icrass 3 Stfa-Boltzma law mk R( T ) εσt σ W / ( 2 4 m K ) Total powr / ara radiatd icrass as T 4 6

7 Attmpts to calculat th spctral distributio of blackbody radiatio from first pricipls faild Th bst dscriptio was giv by th Rayligh-Jas formula 2πckT λ ( λ, T ) 4 I This dscribd th distributio at log wavlgths but icrasd without limit as λ Ultraviolt catastroph 7

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9 Plack was abl to calculat th corrct distributio by assumig rgy was quatizd (h was dsprat) Microscopic (atomic) oscillators ca oly hav crtai discrt rgis E hf h Js Th oscillators ca oly absorb or mit rgy i multipls of ΔE hf 9

10 Plack s radiatio law agrd with data I ( ) λ 2 2πc h 5 λ 1, T hc / λkt 1 It lads dirctly to Wi s displacmt law ad th Stfa-Boltzma law It agrs with Rayligh-Jas formula for larg wavlgths S drivatios for both i Thorto ad Rx 1

11 W rprst a blackbody by a cavity hatd to tmpratur T ad coctd to th outsid by a small hol W ll assum a mtal cavity i th form of a cub (a ov with a pihol) Thrmal agitatio causs th lctros i th wall to oscillat (acclrat) thus producig lctromagtic radiatio Th lctromagtic radiatio forms stadig wavs isid th cavity with ods at th mtallic surfacs 11

12 Th calculatio of Plack s law has fiv parts N(f)df Numbr of stadig wavs with frqucis btw f ad f+df W ll do th o-dimsioal cas ad just writ dow th rsult for th thr-dimsioal cas ε avrag rgy pr stadig wav W ll do th calculatio Divid by th volum Chag variabls from frqucy to wavlgth Multiply by c/4 to chag from rgy/volum/wavlgth to rgy/tim/ara/wavlgth 12

13 W first calculat th umbr of stadig wavs i th frqucy itrval from f to f+df Cosidr a o-dimsioal cavity of lgth a (thik of possibl wavs o a strig with fixd dpoits) Th lctric fild is giv by E( x, t) E si whr fλ c 2πx λ si ( 2πft) So th amplitud has a siusoidal spac variatio which is oscillatig i tim siusoidally 13

14 W wat th amplitud of th lctric fild to vaish at x ad xa At x, this is satisfid automatically At xa, w must hav 2a, λ This dtrmis th allowd valus for th wavlgth What do possibl stadig wavs look lik? 1,2,3,... 14

15 It s a bit asir to work i trms of frqucy f c 2a W rprst th allowd valus of frqucy o a li whr w plot a poit at vry itgral valu of d(2a/c)f W ll us this li to fid N(f)df, th umbr of allowd frqucis (stadig wavs) i th rag f to f+df 15

16 Lookig at th li, th umbr of poits btw f ad f+df is N( f ) df 2 2a c df Whr w multiplid x2 to accout for th two possibl polarizatios of th lctromagtic wav 16

17 Th calculatio for a thr-dimsioal cavity is similar but somwhat mor complicatd W ll just writ dow th rsult 2a 2 N( f ) df π f df c Ad for latr us ot df dλ c 2 λ 3 sic f c λ 17

18 Nxt w calculat th avrag rgy pr stadig wav Classically this is just ε whr εp P P ( ε ) ( ε ) ( ε ) ad w usd dε dε is th Boltzma factor x cx ε kt 1 kt c ε kt ε kt cx 2 ( cx 1) This is th sam rsult th quipartitio thorm givs for two dgrs of frdom dε dε kt 1 kt ε kt 18

19 Kitic Thory of Gass Basd o atomic thory of mattr Rsults iclud Spd of a molcul i a gas v rms (<v 2 >) 1/2 (3kT/m) 1/2 Equipartio thorm Itral rgy U f/2 NkT f/2 RT Hat capacity C V (du/dt) V f/2 R Maxwll spd distributio 19

20 Cotiuig with classical calculatio w hav 3 2 2a 2 1 8πf kt u( f ) df π f df kt 3 3 c a c df Ad chagig variabls from frqucy to wavlgth 8πkT u( λ) dλ dλ 4 λ Multiply by c/4 to fid th Rayligh-Jas formula 2πckT I (, T ) λ 4 λ 2

21 Now rdo th calculatio ala Plack Lt Ehf, th th avrag rgy is ε whr ad 1 kt εp( ε ) P( ε ) ε kt hf kt is hf kt 1 kt hf kt hf kt kt th Boltzma factor P( ε ) 21

22 22 Blackbody Radiatio To valuat this w us stadard tricks from statistical mchaics l d d d d d d

23 23 Blackbody Radiatio Ad ot ( ) X X X X

24 Puttig ths togthr w hav ε ε hf ε d kt l d hf 1 d d l 1 ( ) hf 1 1 hf hf kt hf hf ( 1 ) 1 hc λ d l d 1 ( )( 1 1 ) 1 hc λkt 2 24

25 W alrady kow th umbr of stadig wavs pr volum 8π N( λ) dλ dλ 4 λ So th rgy pr volum is u( λ) dλ hc λ 1 8π dλ λ 1 hc 4 λkt Ad chagig uits to spctral itsity (multiply by c/4) givs Plack s formula 2 2πc h 1 I λ T ( ), 5 λ hc λkt 1 25

26 You ca s how Plack avoidd th ultraviolt catastroph Bcaus th rgy is proportioal to th frqucy Th avrag rgy is kt wh th possibl rgis ar small compard to kt Th avrag rgy is xtrmly small wh th possibl rgis ar larg compard to kt (bcaus P(ε) is xtrmly small) 26

27 Plack s papr is grally cosidrd to b th birthplac of quatum mchaics Rvisioist history? Plack did ot pay too much atttio to rgy quatizatio Nithr did ayo ls Thr is cotrovrsy of whthr h v itdd th rgy of a oscillator to b hf 27

28 Photolctric Effct Light icidt o a mtal will jct lctros Asid, othr mas of doig this ar with tmpratur, lctric filds, ad particl bombardmt 28

29 Photolctric Effct Exprimts showd Th kitic rgy of th photolctros ar idpdt of th light itsity Th maximum kitic rgy of th photolctros dpds o th light frqucy Th smallr th work fuctio φ th smallr th thrshold frqucy to produc photolctros Th umbr of photolctros is proportioal to light itsity Th photolctros ar mittd istataously (o th ordr of aoscods) 29

30 Photolctric Effct 3

31 Photolctric Effct 31

32 Photolctric Effct Explaid by Eisti i o of his aus mirabilis paprs I his papr h assumd Elctromagtic fild was quatizd Light quata wr localizd i spac (lik particls) photos Ergy E hf I th photolctric procss, th rgy quata (photos) ar compltly absorbd 32

33 Photolctric Effct Thus photos ptrat th surfac of th mtal ad ar absorbd by lctros Th lctros ovrcom attractiv forcs that ormally hold thm i th matrial ad scap Cosrvatio of rgy givs hf 1/2mv 2 max + φ Ad cosqutly h prdictd 1/2mv 2 max V hf φ Not h/ ca b masurd from th slop 33

34 Photolctric Effct This was strag sic it ivolvd Plack s costat h This was a difficult xprimt to carry out It took almost a dcad to vrify Millika was th pricipl xprimtr (who was tryig to prov Eisti s thory wrog) Th d rsult was proof that light rgy is quatizd ad Ehf 34

35 Photolctric Effct 35