Lecture #2: Wave Nature of the Electron and the Internal Structure of an Atom

Transcription

1 5.61 Fall 013 Lctur # pag 1 Lctur #: Wav Natur of t Elctro ad t Itral Structur of a Atom Last tim: Surpris Ligt as particl 1. Potolctric ffct, spcially KE vs. ν. Ligt as packts of rgy, calld potos, E = ν.. Compto scattrig Ligt as avig bot KE (scalar) ad p (vctor). Billiard-lik scattrig of poto by Today: 1. Aotr surpris wav caractr of compar diffractio of X-ray ad by sam Al foil masur λ = p dbrogli postulatd, for objcts cosidrd to b particls (.g. ) tat λ = p. Itral structur of mattr: mostly mpty spac! I ordr to fill spac Rutrford postulatd platary atom * radiativ collaps (a fatal flaw) Bor modl * postulatd quatizd (ad cosrvd) agular momtum f = * Explaid li spctrum of H * radiativ collaps avoidd d Brogli * itgr # of λ aroud Bor orbit to prvt aiilatio of t by dstructiv itrfrc X-ray ad diffractio off of Al foil rvisd 9/9/13 9:9:00 AM

2 5.61 Fall 013 Lctur # pag OR X-ray Bam bam Bam stop targt fluorsct Al foil scr Us kow lattic spacigs i Al as a rulr to masur λ of S Figur 1.13 i McQuarri, pag 30. Obtai bull s y diffractio pattr for bot X-ray ad. Pairs of rigs com from scattrig off arst ad scod arst igbor atoms. Ca coos λ of X-rays (masurd i sparat xprimt). p, p = m q V 1/. Ca cotrol p of : E = q V = m W t rig pattrs matc, must av t sam λ as X-rays (λ is a way to masur λ, w fid tat it is λ =. p small p, larg λ larg p, small λ x-ray =λ ). Tis How dos diffractio work? No-Lctur 1/ a a a is uit cll distac a rvisd 9/9/13 9:9:00 AM

3 5.61 Fall 013 Lctur # pag 3 S pag 59 of Karplus ad Portr. λ icidt bam (pla wav) θ θ θ diffractd bam a Gt costructiv itrfrc from scattrig off of atoms sparatd by a w t agl θ btw icidt ad dtctd radiatio is suc tat t pats diffr by λ. ( is a itgr, t ordr ) Suppos λ = a/ arst igbor a si θ arst igbor a θ d arst igbor 1/ a si θ 1/ a λ = asiθ 1 st ordr w = 1 λ = 1/ asiθ λ θ = si 1 1 st ordr w = 1 a d arst igbor θ = si 1 θ = si 1 3/ = 1 θ 1 = 0.5 θ 1 = 0.36 = θ = 1.57 θ = 0.79 θ = si 1 θ λ 1/ a T 1 st -ordr rig for t scod arst igbor is of smallr diamtr ta tat for t arst igbor. rvisd 9/9/13 9:9:00 AM

4 5.61 Fall 013 Lctur # pag 4 T agular sparatio btw rig pairs is largr i scod ordr. [Tis rmids you tat a gratig opratd i scod ordr givs igr spctral rsolutio ta i first ordr.] Diffrt rig pattrs (spacigs ad rlativ itsitis) ar obsrvd for simpl cubic fac-ctrd cubic body-ctrd cubic xagoal clos packd SC FCC BCC HCP Ca us powdr pattr to assig uit cll typ ad to dtrmi t uit cll distacs. Wy is it calld it a powdr pattr? END of NON-LECTURE So w ca masur a usig X-rays of kow λ. W ca t masur λ of for masurd p (masurd i a sparat xprimt). Fid mpirically tat λ = p. Eisti (1906 spcial rlativity) sowd, for ligt E ν p = = =. c λν λ DBrogli, i is 194 P.D. Tsis, postulatd tat if p = for ligt, t t sam λ formula would giv λ for particls. Was tis just a good guss or a dp isigt? Exampl calculatio ad a paradox For a, if λ = 1Å, t v =? E =? Typical diamtr of a atom is ~1Å. Wat E or v of a orbitig lctro would b rquird for a to liv isid a atom? J s λ = m = = Tus, for λ = 1Å, w prdict p ( ). 31 kg)(v[m / s] v = m / s.4% of c (spd of ligt). E = 1 mv = J = ( V / J )( J ) (uits covrsio) = 149V 10 typical ioizatio rgy! (Ergy rquird to rmov o from a gas pas atom or molcul.) rvisd 9/9/13 9:9:00 AM

5 5.61 Fall 013 Lctur # pag 5 Paradox: How ca fit isid of a atom? Typical diamtr of a atom is ~1Å. Typical bidig rgy of to uclus is 15V. Yt w rquir tat, for a to av small oug λ to fit isid a atom, it must av a rgy 10 too larg to rmai trappd isid t atom. Tis is a paradox. To udrstad tis w must go back to t qustio: wat is t itral structur of a atom? Erst Rutrford 1911 α-particl scattrig off a mtal foil. Exprimts by Has Gigr ad Erst Marsd. α-particl (H + + io io) back-scattr dtctor scr ti foil forward-scattr dtctor dtctor scr scr Foud tat vry fw α-particls wr back-scattrd. If atoms wr omogous ad w kow t # of atoms pr ara of foil (ow would w kow tis?), w kow tat most of t α particls sould arriv at t dtctor scr udflctd or oly wakly. Fractio backscattrd tlls us t ffctiv diamtr, mass, ad carg of ac atom, as s by a α particl. Foud t atomic diamtr dtrmid i tis way to b vastly smallr ta t distac btw atoms (calculatd from t masurd dsity of t mtal of t foil ad its kow atomic wigt). Jllium (omogous atom) is ruld out. Most of mass of atom is localizd isid a vry small diamtr uclus. Yt atoms ar spac fillig. [Also, t carg o tis uclus is positiv ad qual to t atomic umbr of t atom.] rvisd 9/9/13 9:9:00 AM

6 5.61 Fall 013 Lctur # pag 6 How do w ratioaliz ts two smigly cotradictory facts? Platary modl of a atom proposd by Rutrford: orbit + uclus Coulomb attractio btw uclus ad : Ctrifugal forc: F i = F out, solv for v F q 1 r = i SI uits, ε 0 is prmittivity (1) 4πε 0 of fr spac m v F out =+ () r q 1 1/ v = 4πε 0 m r. (3) W av (tratig uclus as ifiitly avy), for vry valu of t circular orbit radius, r, a costat spd, v, alog t orbit. No quatizatio of r, v, or E is rquird, but small r must corrspod to larg v. W av a acclratd carg radiats rgy = ν wr ν is t frqucy of t orbital motio. Loss rgy. Must mov to smallr radius orbit. Orbit frqucy icrass. Rat of radiatio of rgy icrass. Radiativ collaps. Bad! Ca tis cotradictio b xplaid? rvisd 9/9/13 9:9:00 AM

7 5.61 Fall 013 Lctur # pag 7 circumfrc = π r π r priod = v v q 1 frqucy= = 3 πr rπε 0 m 4π r small r, ig frqucy spirals i toward uclus. No-Lctur ν t To avoid tis fatal flaw, Bor postulatd tat lctro agular momtum, f, is quatizd, ad trfor, cosrvd. f = r p = m vr = (4) v = r p m r Tis provids a artificial aswr to t problm of radiativ collaps. Wy artificial? Combi Eqs. (3) ad (4) to solv for t orbit radius: Orbit radius, r, gts larg as 4πε r 0 = = [0.59 q 059 ]Å m f atomic uit of lgt (or ) icrass. Istad, solv for t orbit vlocity: 1 q 1 v = = [ ]m / s 4πε 0 v gts small as f (or ) icrass. Nxt, solv for λ : 1 q m p = 4πε 0 8π ε 0 π r λ = = = = [3.3]Å p q m Circumfrc of t orbit = πr = λ! Tis is a plasat surpris. T fits isid a atom if its wavlgt is arragd alog t orbit circumfrc! Nxt, solv for t allowd rgy lvls: rvisd 9/9/13 9:9:00 AM

8 5.61 Fall 013 Lctur # pag 8 1 q 1 E = T r ( )+ V(r ) = m v 4πε 0 r 4 m q 1 = 8ε 0 I I Rydbrg costat for H cr H Balmr, from umrical study of t missio lis of H atom, proposd tat t frqucy 1 of a spctral li is giv by tims t diffrc btw rgy lvls. Trasitios btw discrt (quatizd) rgy lvls! ' Turs out tat ν ', = E ' E * Tis rgy lvl formula for t Bor atom xactly rproducs all of t rgy m m uclus lvls of all 1 systms rplacig m by μ = m m + m uclus * accouts for li spctra if ν =, E E * accordig to Bor t lowst possibl valu of f is f = 1 (w will fid out latr tat f = 0 is also possibl) * dos ot accout for rlativ itsitis of spctral lis, or radiativ liftims of rgy lvls * dos ot xplai ffct of a magtic fild o spctrum * mor satisfyig to us dbrogli s ida to prvt radiativ collaps. But isistc o itgr orbital agular momta [Bor] ad avoidac of dstructiv itrfrc aroud a circular orbit [d Brogli] ar bot artificial idas. * if t is movig o a circular orbit, it must radiat, but is it rally movig? WEIRD! dbrogli: Itgr umbr of λ aroud orbit circumfrc is wat is rquird to prvt dstructiv itrfrc of wit itslf. A sould ot disappar. Sms mor fudamtal. But w ar just makig ad oc proposals to xplai a fudamtal cotradictio. W sould ot b comfortabl wit ay of tis! Bor tory caot xplai ay spctra otr ta 1 spctra. At tis stag, t spctrum of t simplst possibl atom, H, rmais a mystry. rvisd 9/9/13 9:9:00 AM

9 5.61 Fall 013 Lctur # pag 9 Wat do w do ow? Nxt Lctur: Discuss wav quatio i prparatio for Scrödigr Equatio, wic is a mor complt ad widly applicabl way to dal wit ts uxpctd rsults. rvisd 9/9/13 9:9:00 AM

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