Chapter 4. Problem Solutions

Transcription

1 Chapter 4. Prblem Slutis. The great majrity f alpha particles pass thrugh gases ad thi metal fils with deflectis. T what cclusi abut atmic structure des this bservati lead? The fact that mst particles pass thrugh udetected meas that there is t much t deflect these particles; mst f the vlume f a atm is empty space, ad gases ad metals are verall electrically eutral. 3. Determie the distace f clsest apprach f.00-mev prts icidet gld uclei. Fr a "clsest apprach", the icidet prt must be directed "head-" t the ucleus, with agular mmetum with respect t the ucleus (a "Impact parameter" f zer; see the Appedix t Chapter 4). I this case, at the pit f clsest apprach the prt will have kietic eergy, ad s the ptetial eergy at clsest apprach will be the iitial kietic eergy, takig the ptetial eergy t be zer i the limit f very large separati. Equatig these eergies, r mi 4πε Ze K iitial K iitial ( N m / C Ze 4πε r mi, r ( 79)( ) J C) m.

2 5. What is the shrtest wavelegth preset i the Brackett series f spectral lies? The wavelegths i the Brackett series are give i Equati (4.9); the shrtest wavelegth (highest eergy) crrespds t the largest value f. Fr, λ m.46 µ m R m 7. I the Bhr mdel, the electr is i cstat mti. Hw ca such a electr have a egative amut f eergy? While the kietic eergy f ay particle is psitive, the ptetial eergy f ay pair f particles that are mutually attracted is egative. Fr the system t be bud, the ttal eergy, the sum f the psitive kietic eergy ad the ttal egative ptetial eergy, must be egative. Fr a classical particle subject t a iverse-square attractive frce (such as tw ppsitely charged particles r tw uifrm spheres subject t gravitatial attracti i a circular rbit, the ptetial eergy is twice the egative f the kietic eergy.

3 9. The fie structure cstat is defied as α e /ε hc. This quatity gt its ame because it first appeared i a thery by the Germa physicist Arld Smmerfeld that tried t explai the fie structure i spectral lies (multiple lies clse tgether istead f sigle lies) by assumig that elliptical as well as circular rbits are pssible i the Bhr mdel. Smmerfeld's apprach was the wrg track, but α has evertheless tured ut t be a useful quatity i atmic physics. (a) Shw that α v /c, where v, is the velcity f the electr i the grud state f the Bhr atm. (b) Shw that the value f α is very clse t /37 ad is a pure umber with dimesis. Because the magetic behavir f a mvig charge depeds its velcity, the small value f α is represetative f the relative magitudes f the magetic ad electric aspects f electr behavir i a atm. (c) Shw that αa λ c /π, where a is the radius f the grud-state Bhr rbit ad λ c is the Cmpt wavelegth f the electr. (a) The velcity v, is give by Equati (4.4), with r r a. Cmbiig t fid v, v (b) Frm the abve, α e 4πε ma e ε πε h 4 m πme 4 e 4ε h v e, s α. c ε h c 9 ( 60 0 C) 34 8 ( C / N m )( J s)( m/s) ,

4 s that /α 37. t fur sigificat figures. A clse cheek f the uits is wrthwhile; treatig the uits as algebraic quatities the uits as give i the abve calculati are [C ] [N][m [C ] [m] [ J][ s] ] [s] [N m] [J]. Thus, α is a dimesiless quatity, ad will have the same umerical value i ay system f uits. The mst accurate (Nvember, 00) value f /α is , α accurate t better tha 4 parts per billi. (c) Usig the abve expressi fr α ad Equati (4.3) with fr a, e h ε α h a ε hc πme π mc where the Cmpt wavelegth λ C is give by Equati (.). λc π,

5 . Fid the quatum umber that characterizes the earth's rbit arud the su. The earth's mass is 6.0 x 0 4 kg, its rbital radius is.5 x 0 m, ad its rbital speed is 3.0 x 0 4 m/s. With the mass, rbital speed ad rbital radius f the earth kw, the earth's rbital agular mmetum is kw, ad the quatum umber that wuld characterize the earth's rbit abut the su wuld be this agular mmetum divided by ; 4 4 L mvr ( kg)(3.0 0 m/s)(.5 0 m) h h J s (The umber f sigificat figures t f ccer.) 3. Cmpare the ucertaity i the mmetum f a electr cfied t a regi f liear dimesi a with the mmetum f a electr i a grud-state Bhr rbit. The ucertaity i psiti f a electr cfied t such a regi is, frm Equati (3.), p > /a, while the magitude f the liear mmetum f a electr i the first Bhr rbit is h h h p ; λ πa a the value f p fud frm Equati (3.3) is half f this mmetum.

6 5. What effect wuld yu expect the rapid radm mti f the atms f a excited gas t have the spectral lies they prduce? The Dppler effect shifts the frequecies f the emitted light t bth higher ad lwer frequecies t prduce wider lies tha atms at rest wuld give rise t. 7. A prt ad a electr, bth at rest iitially, cmbie t frm a hydrge atm i the grud state. A sigle pht is emitted i this prcess. What is its wavelegth? It must assumed that the iitial electrstatic ptetial eergy is egligible, s that the fial eergy f the hydrge atm is E -3.6 ev. The eergy f the pht emitted is the -E l, ad the wavelegth is hc λ E ev m 3. 6 ev 9. 0 m 9. m, i the ultravilet part f the spectrum (see, fr istace, the back edpapers f the text).

7 9. Fid the wavelegth f the spectral lie that crrespds t a trasiti i hydrge frm the 0 state t the grud state. I what part f the spectrum is this? Frm either Equati (4.7) with 0 r Equati (4.8) with f ad i 0, λ 9. 0 m 9. m, 99 R m which is i the ultravilet part f the spectrum (see, fr istace, the back edpapers f the text).. A beam f electrs bmbards a sample f hydrge. Thrugh what ptetial differece must the electrs have bee accelerated if the first lie f the Balmer series is t be emitted? The electrs eergy must be at least the differece betwee the ad 3 levels, 8 E E 3 E E ( 3. 6 ev). ev 9 9 (this assumes that few r e f the hydrge atms had electrs i the level). A ptetial differece f. ev is ecessary t accelerate the electrs t this eergy.

8 3. The lgest wavelegth i the Lyma series is.5 m ad the shrtest wavelegth i the Balmer series is m. Use the figures t fid the lgest wavelegth f light that culd iize hydrge. The eergy eeded t iize hydrge will be the eergy eeded t raise the eergy frm the grud state t the first excited state plus the eergy eeded t iize a atm i the secd excited state; these are the eergies that crrespd t the lgest wavelegth (least eergetic pht) i the Lyma series ad the shrtest wavelegth (mst eergetic pht) i the Balmer series. The eergies are prprtial t the reciprcals f the wavelegths, ad s the wavelegth f the pht eeded t iize hydrge is λ λ λ As a check, te that this wavelegth is R m m 9. 3m. 5. A excited hydrge atm emits a pht f wavelegth λ i returig t the grud state. (a) Derive a frmula that gives the quatum umber f the iitial excited state i terms f λ ad R. (b) Use this frmula t fid i fr a 0.55-m pht. (a) Frm Equati (4.7) with i, R, λ i which is slved fr

9 i / λr λr. λr (b) Either f the abve frms gives very clse (fur place) t 3; specifically, with the prduct λr (0.55x0-9 m)(.097x0 7 m - ).5 ruded t fur places as 9/8, 3 exactly. 7. Whe a excited atm emits a pht, the liear mmetum f the pht must be balaced by the recil mmetum f the atm. As a result, sme f the excitati eergy f the atm ges it the kietic eergy f its recil. (a) Mdify Eq. (4.6) t iclude this effect. (b) Fid the rati betwee the recil eergy ad the pht eergy fr the 3 trasiti i hydrge, fr which E f - E i.9 ev. Is the effect a majr e? A relativistic calculati is sufficiet here. (a) A relativistic calculati wuld ecessarily ivlve the chage i mass f the atm due t the chage i eergy f the system. The fact that this mass chage is t small t measure (that is, the chage is measured idirectly by measurig the eergies f the emitted phts) meas that a relativistic calculati shuld suffice. I this situati, the kietic eergy f the recilig atm is p ( hν / c) K, M M where m is the ftequecy f the emitted pht ad p h/λ hν/c is the magitude f the mmetum f bth the pht ad the recilig atm. Equati (4.6) is the

10 ( hν) hν Ei E f hν K hν hν. Mc Mc This result is equivalet t that f Prblem -53, where hν E. ad the term p /(M) crrespds t E - E i that prblem. As i Prblem -53, a relativistic calculati is maageable; the result wuld be Mc E f Ei hν, hν a frm t fte useful; see part (b). (b) As idicated abve ad i the prblem statemet, a relativistle calculati is sufficiet. As i part (a), ( E c ) p / K E. 9 ev 9 K, ad. 0 0, M M E 6 Mc ( ev) r.0 x 0-9 t tw sigificat figures. I the abve, the rest eergy f the hydrge atm is frm the frt edpapers.

11 9. Shw that the frequecy f the pht emitted by a hydrge atm i gig frm the level t the level is always itermediate betwee the frequecies f revluti f the electr i the respective rbits. There are may equivalet algebraic methds that may be used t derive Equati (4.9), ad that result will be cited here;. 3 h E f The frequecy v f the pht emitted i gig frm the level t the level is btaied frm Equati (4.7) with i ad f ;. ) ( ) ( h E h E ν This ca be see t be equivalet t the expressi fr v i terms f ad p that was fud i the derivati f Equati (4.0), but with replaced by ad p. Nte that i this frm, ν is psitive because E l is egative. Frm this expressi f, f h E < 3 ν as the term i brackets is less tha. Similarly,

12 ν ( )( ) ( E )( ) 3 f h( ) > f as the term i brackets is greater tha., 3. A µ - mu is i the state f a muic atm whse ucleus is a prt. Fid the wavelegth f the pht emitted whe the muic atm drps t its grud state. I what part f the spectrum is this wavelegth? Fr a muic atm, the Rydberg cstat is multiplied by the rati f the reduced masses f the muic atm ad the hydrge atm, R' R (m'/m e ) 86R, as i Example 4.7; frm Equati (4.7), 4/ 3 4/ 3 0 λ m 0.653m, R 7-86( m ) i the x-ray rage. µ p µ 07 me, m p me m 86me mµ m p m 836 m m

13 33. A mixture f rdiary hydrge ad tritium, a hydrge istpe whse ucleus is apprximately 3 times mre massive tha rdiary hydrge, is excited ad its spectrum bserved. Hw far apart i wavelegth will the H α lies f the tw kids f hydrge be? The H α lies, crrespdig t 3 i Equati (4.6), have wavelegths f λ (36/5) (/R). Fr a tritium atm, the wavelegth wuld be λ T (36/5) (/RT), where RT is the Rydberg cstat evaluated with the reduced mass f the tritium atm replacig the reduced mass f the hydrge atm. The differece betwee the wavelegths wuld the be λ T R λ λ λt λ λ λ. RT The values f R ad RT are prprtial t the respective reduced masses, ad their rati is R memh /( me mh ) mh( me mt ). RT memt /( me mt ) mt( me mh ) Usig this i the abve expressi fr λ, me( mt mh ) m λ λ e λ, me( me mh ) 3m H where the apprximatis m e r H m H ad m T 3m H have bee used. Isertig umerical values, 3 ( 36/ 5) ( 9. 0 kg) 0 λ ( m ) 3 ( kg) m 0.38 m.

14 35. (a) Derive a frmula fr the eergy levels f a hydrgeic atm, which is a i such as He r Li whse uclear charge is Ze ad which ctais a sigle electr. (b) Sketch the eergy levels f the He' i ad cmpare them with the eergy levels f the H atm. (c) A electr jis a bare helium ucleus t frm a He i. Fid the wavelegth f the pht emitted i this prcess if the electr is assumed t have had kietic eergy whe it cmbied with the ucleus. (a) The steps leadig t Equati (4.5) are repeated, with Ze istead f e ad Z e 4 istead f e 4, givig 4 m Z e E, 8πε h where the reduced mass m' will deped the mass f the ucleus. (b) A plt f the eergy levels is give belw. The scale is clse, but t exact, ad f curse there are may mre levels crrespdig t higher. I the apprximati that the reduced masses are the same, fr He, with Z, the level is the same as the level fr Hydrge, ad the 4 level is the same as the level fr hydrge.

15 The eergy levels fr H ad He : (c) Whe the electr jis the Helium ucleus, the electr-ucleus system lses eergy; the emitted pht will have lst eergy E 4 (-3.6 ev) ev, where the result f part (a) has bee used. The emitted pht's wavelegth is λ hc E ev m ev. 8 0 m.8 m.

16 37. A certai ruby laser emits.00-j pulses f light whse wavelegth is 694 m. What is the miimum umber f Cr 3 is i the ruby? The miimum umber f Cr 3 is will he the miimum umber f phts, which is the ttal eergy f the pulse divided by the eergy f each pht, E Eλ hc / λ hc -9 (. 00 J)(694 0 m) is ( J s)(3.0 0 m/s) 39. The Rutherfrd scatterig frmula fails t agree with the data at very small scatterig agles. Ca yu thik f a reas? Small agles crrespd t particles that are t scattered much at all, ad the structure f the atm des t affect these particles. T these peetratig particles, the ucleus is either partially r cmpletely screeed by the atm's electr clud, ad the scatterig aalysis, based a pitlike psitively charged ucleus, is t applicable.

17 4. A 5.0-MeV alpha particle appraches a gld ucleus with a impact parameter f.6 x 0-3 m. Thrugh what agle will it be scattered? Frm Equati (4.9), usig the value fr /4πε give i the frt edpapers, ct θ -3 ( 5. 0 ev)(.60 0 J/MeV) 9 ( N m / C )( 79)( keepig extra sigificat figures. The scatterig agle is the θ ct (. 43) ta C). 43 ( m).43, 43. What fracti f a beam f 7.7-MeV alpha particles icidet up a gld fil 3.0 x 0-7 m thick is scattered by less tha? The fracti scattered by less tha is - f, with f give i Equati (4.3);

18 f Ze πt 4 πεk π( ct m -3 )( ( 79)(. 6 0 ( 7. 7MeV)(.6 0 θ πt 4πε C) J/MeV) m)(9.0 0 Ze K ct 9 ( 0. 5 ct N m ) θ / C 0. 6, where, the umber f gld atms per uit vlume, is frm Example 4.8. The fracti scattered by less tha is - f ) 45. Shw that twice as may alpha particles are scattered by a fil thrugh agles betwee 60 ad 90 as are scattered thrugh agles f 90 r mre. Regardig f as a fucti f 0 i Equati (4.3), the umber f particles scattered betwee 60 ad 90 is f (60 ) - f (90 ), ad the umber scattered thrugh agles greater tha 90 is just f (90 ), ad f ( 60 ) f ( 90 ) ct ( 30 ) ct ( 45 ) 3, f ( 90 ) ct ( 45 ) s twice as may particles are scattered betwee 60 ad 90 tha are scattered thrugh agles greater tha 90.

19 47. I special relativity, a pht ca be thught f as havig a mass f m E ν /c. This suggests that we ca treat a pht that passes ear the su i the same way as Rutherfrd treated a alpha particle that passes ear a ucleus, with a attractive gravitatial frce replacig the repulsive electrical frce. Adapt Eq. (4.9) t this situati ad fid the agle f deflecti θ fr a pht that passes b R su frm the ceter f the su. The mass ad radius f the su are respectively.0 x 0 30 kg ad 7.0 x 0 8 m. I fact, geeral relativity shws that this result is exactly half the actual deflecti, a cclusi supprted by bservatis made durig slar clipses as metied i Sec..0. If gravity acted phts as if they were massive bjects with mass m E v /c, the magitude f the frce F i Equati (4.8) wuld be GM m F su ; r the factrs f r wuld cacel, as they d fr the Culmb frce, ad the result is θ θ θ c b mc b si GMsum cs ad ct GM a result that is idepedet f the pht s eergy. Usig b R su, θ ta GM c R 4 su su deg ta su ( N m / kg )( (3.0 0 m/s)(7.0 0 m), 30 kg)