Structure of the Atom. Thomson s Atomic Model. Knowledge of atoms in Experiments of Geiger and Marsden 2. Experiments of Geiger and Marsden

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1 CHAPTER 4 Structur of th Atom 4.1 Th Atomic Modls of Thomson and Ruthrford 4. Ruthrford Scattring 4.3 Th Classic Atomic Modl 4.4 Th Bohr Modl of th Hydrogn Atom 4.5 Succsss & Failurs of th Bohr Modl 4.6 Charactristic X-Ray Spctra and Atomic Numbr 4.7 Atomic Excitation by Elctrons Th opposit of a corrct statmnt is a fals statmnt. But th opposit of a profound truth may wll b anothr profound truth. An xprt is a prson who has mad all th mistaks that can b mad in a vry narrow fild. Nvr xprss yourslf mor clarly than you ar abl to think. Prdiction is vry difficult, spcially about th futur. Nils Bohr ( ) - Nils Bohr Structur of th Atom Evidnc in 19 indicatd that th atom was not a fundamntal unit: 1) Thr smd to b too many kinds of atoms, ach blonging to a distinct chmical lmnt (way mor than arth, air, watr, and fir!). ) Atoms and lctromagntic phnomna wr intimatly rlatd (magntic matrials; insulators vs. conductors; diffrnt mission spctra). 3) Elmnts combin with som lmnts but not with othrs, a charactristic that hintd at an intrnal atomic structur (valnc). 4) Th discovris of radioactivity, x rays, and th lctron (all smd to involv atoms braking apart in som way). Knowldg of atoms in 19 Thomson s Atomic Modl Elctrons (discovrd in 1897) carrid th ngativ charg. Elctrons wr vry light, vn compard to th atom. Protons had not yt bn discovrd, but clarly positiv charg had to b prsnt to achiv charg nutrality. Thomson s plum-pudding modl of th atom had th positiv chargs sprad uniformly throughout a sphr th siz of th atom, with lctrons mbddd in th uniform background. In Thomson s viw, whn th atom was hatd, th lctrons could vibrat about thir quilibrium positions, thus producing lctromagntic radiation. Unfortunatly, Thomson couldn t xplain spctra with this modl. Exprimnts of Gigr and Marsdn Ruthrford, Gigr, and Marsdn concivd a nw tchniqu for invstigating th structur of mattr by scattring α particls from atoms. Exprimnts of Gigr and Marsdn Gigr showd that many α particls wr scattrd from thin gold-laf targts at backward angls gratr than 9. 1

2 Elctrons can t backscattr α particls. Bfor Aftr Calculat th maximum scattring angl corrsponding to th maximum momntum chang. Try multipl scattring from lctrons If an α particl is scattrd by N lctrons: N th numbr of atoms across th thin gold layr, t m: n It can b shown that th maximum momntum transfr to th α particl is: p m v α max Dtrmin θ max by ltting p max b prpndicular to th dirction of motion: Th distanc btwn atoms, d n -1/3, is: θ max pα mvα too small! p M v α α α N t / d still too small! Ruthrford s Atomic Modl 4.: Ruthrford Scattring vn if th α particl is scattrd from all 79 lctrons in ach atom of gold. Exprimntal rsults wr not consistnt with Thomson s atomic modl. Ruthrford proposd that an atom has a positivly chargd cor (nuclus) surroundd by th ngativ lctrons. Gigr and Marsdn confirmd th ida in Ernst Ruthrford ( ) Scattring xprimnts hlp us study mattr too small to b obsrvd dirctly. Thr s a rlationship btwn th impact paramtr b and th scattring angl θ. Whn b is small, r is small. th Coulomb forc is larg. θ can b larg and th particl can b rplld backward. whr K 1 mv cot(θ/) π θ Th cross sction σ π b is rlatd to th probability for a particl bing scattrd by a nuclus (t foil thicknss): Th fraction of incidnt particls scattrd is: Th numbr of scattring nucli pr unit ara: Ruthrford Scattring Any particl insid th circl of ara π b will b similarly (or mor) scattrd. Ruthrford Scattring Equation Th numbr of particls scattrd pr unit ara is: In actual xprimnts, a dtctor is positiond from θ to θ + dθ that corrsponds to incidnt particls btwn b and b + db.

3 Ruthrford scattring xprimnt 1 MV protons scattring off gold foil. Not th corrct dpndnc on scattring angl. 4.3: Th Classical Atomic Modl Considr an atom as a plantary systm. Th Nwton s nd Law forc of attraction on th lctron by th nuclus is: 1 mv F r r whr v is th tangntial vlocity of th lctron: v 1 1 K mv mr r Th total nrgy is thn: This is ngativ, so th systm is bound, which is good. Th Plantary Modl is Doomd 4.4: Th Bohr Modl of th Hydrogn Atom From classical E&M thory, an acclratd lctric charg radiats nrgy (lctromagntic radiation), which mans th total nrgy must dcras. So th radius r must dcras!! Elctron crashs into th nuclus!? Bohr s gnral assumptions: 1. Stationary stats, in which orbiting lctrons do not radiat nrgy, xist in atoms and hav wll-dfind nrgis, E n. Transitions can occur btwn thm, yilding light of nrgy: E E n E n hν. Classical laws of physics do not apply to transitions btwn stationary stats, but thy do apply lswhr. n n 1 n 3 Physics had rachd a turning point in 19 with Planck s hypothsis of th quantum bhavior of radiation, so a radical solution would b considrd possibl. 3. Th angular momntum of th n th stat is: nħ whr n is calld th Principal Quantum Numbr. Angular momntum is quantizd! Consquncs of th Bohr Modl Bohr Radius Th angular momntum is: But: v Solving for r n : L mvr nħ So th vlocity is: v n ħ / mr mr rn So: n a nħ m r mr whr: a ħ a m Th Bohr radius, ħ a m is th radius of th unxcitd hydrogn atom and is qual to: / Th ground stat Hydrogn atom diamtr is: a is calld th Bohr radius. It s th diamtr of th Hydrogn atom (in its lowst-nrgy, or ground, stat). 3

4 Th Hydrogn Atom Enrgis Us th classical rsult for th E nrgy: 8πε r and: r nħ m n So th nrgis of th stationary stats ar: Th Hydrogn Atom Emission of light occurs whn th atom is in an xcitd stat and dcays to a lowr nrgy stat (n u n l ). 1 ν hν λ c hc hν Eu E l whr ν is th frquncy of a photon. or: E n E /n whr E 13.6 V. R is th Rydbrg constant. R 4 m 3 (4 πħ) cε Transitions in th Hydrogn Atom Th atom will rmain in th xcitd stat for a short tim bfor mitting a photon and rturning to a lowr stationary stat. In quilibrium, all hydrogn atoms xist in n : Charactristic X-Ray Spctra and Atomic Numbr Shlls hav lttr nams: K shll for n 1 L shll for n Th atom is most stabl in its ground stat. An lctron from highr shlls will fill th innr-shll vacancy at lowr nrgy. Whn it occurs in a havy atom, th radiation mittd is an x-ray. It has th nrgy E (x-ray) E u E l. Atomic Numbr and Mosly Th x-rays hav nams: L shll to K shll: K α x-ray M shll to K shll: K β x-ray tc. K β K α Mosly s Empirical Rsults Th K α x-ray is producd from th n to n 1 transition. In gnral, th K sris of x-ray wavlngths ar: G.J. Mosly studid x-ray mission in Atomic numbr Z numbr of protons in th nuclus. Mosly found a rlationship btwn th frquncis of th charactristic x-ray and Z. Mosly found this rlation holds for th K α x-ray: ν Kα 3cR ( Z 1) 4 W us Z-1 instad of Z bcaus on lctron is alrady prsnt in th K-shll and so shilds th othr(s) from th nuclus charg. Mosly s rsarch clarifid th importanc of Z and th lctron shlls for all th lmnts, not just for hydrogn. 4

5 Th Corrspondnc Principl Bohr s corrspondnc principl is rathr obvious: Th Corrspondnc Principl Th frquncy of th radiation mittd ν classical is qual to th orbital frquncy ν orb of th lctron around th nuclus. ν classical ω ν v / r ν orb classical π π This should agr with th frquncy of th transition from n + 1 to n (whn n is vry larg): ν Bohr In th limits whr classical and quantum thoris should agr, th quantum thory must rduc th classical rsult. For larg n: ν Bohr Substituting for E : ν Bohr ν classical 4.7: Atomic Excitation by Elctrons Franck and Hrtz studid th phnomnon of ionization. Atomic Excitation by Elctrons Ground stat has E to b zro. First xcitd stat has E 1. Th nrgy diffrnc E 1 E 1 is th xcitation nrgy. Hg has an xcitation nrgy of 4.88 V in th first xcitd stat No nrgy can b transfrrd to Hg blow 4.88 V bcaus not nough nrgy is availabl to xcit an lctron to th nxt nrgy lvl Acclrating voltag is blow 5 V: lctrons did not los nrgy. Acclrating voltag is abov 5 V: suddn drop in th currnt. Abov 4.88 V, th currnt drops bcaus scattrd lctrons no longr rach th collctor until th acclrating voltag rachs 9.8 V and so on. Fin Structur Constant Th lctron s vlocity in th Bohr modl: v n In th ground stat, v m/s ~ 1% of th spd of light. 4.5: Succsss and Failurs of th Bohr Modl Succss: Th lctron and hydrogn nuclus actually rvolv about thir mutual cntr of mass. Th lctron mass is rplacd by its rducd mass: Th ratio of v 1 to c is th fin structur constant. v α 1 c Th Rydbrg constant for infinit nuclar mass, R, is rplacd by R. 5

6 Limitations of th Bohr Modl Th Bohr modl was a grat stp in th nw quantum thory, but it had its limitations. Failurs: Works only for singl-lctron ( hydrognic ) atoms. Could not account for th intnsitis or th fin structur of th spctral lins (for xampl, in magntic filds). Could not xplain th binding of atoms into molculs. 6