Lecture 36 (Atomic Spectra) Physics 6-1 Sprig 18 Douglas Fields
Frauhofer Lies I the late 17s ad early 18s, oe of the premier skills was that of glassmaker. Joseph Frauhofer became oe of the most skilled ad sought after glassmakers. He discovered a large set of missig colors i the solar spectrum while lookig for a bright lie istead. Thus i 1814, Frauhofer iveted the spectroscope. I the course of his experimets he discovered the bright fixed lie which appears i the orage color of the spectrum whe it is produced by the light of fire. This lie eabled him afterward to determie the absolute power of refractio i differet substaces. Experimets to ascertai whether the solar spectrum cotaied the same bright lie i the orage as that produced by the light of fire led him to the discovery of 574 dark fixed lies i the solar spectrum; millios of such fixed absorptio lies are ow kow. - Wikipedia
He missed a few
Emissio Lies About the same time, people were usig flame emissio spectroscopy to see that differet materials emitted light of specific frequecies whe heated to high eough temperatures that they became heated gases.
Emissio Lies Aother way to do this is to pass a electric curret through a gas, thermally heatig it ad seeig the spectra of light emissio usig a diffractio gratig.
Absorptio & Emissio The other aspect that was oticed is that spectra emitted by a heated elemet was the same as the spectra of absorptio whe cotiuous light was passed through the same cool elemet (as a gas). This cofirmed that the spectra were a property of the elemet. Blackbody radiator Heated gas Blackbody radiatio passed through the same cool gas
Hydroge Eergy Levels I 1885, Joha Balmer discovered a patter for the visible emissio spectra of hydroge. 41,434, 486, ad 656 m His formula was: B, B 364.5 m, Later, Rydberg reformulated the Balmer formula as: 1 4 1 1 1 1 R H for 3,4,5,... B R H 1,973,731.57 m 1
Thomso Model I 194, J.J. Thomso proposed that atoms were composed of a amorphous positively charged plum puddig with embedded electros. The absorptio ad emissio of light could the be accouted for by these electros havig certai atural frequecies of harmoic oscillatio. Oe could imagie rigs of electros with differet paths, or sprigs, or some such cofiguratio which would led each electro with a atural frequecy of oscillatio, ad hece absorb ad emit light with those frequecies.
Gold Foil Experimet I 199, Has Geiger ad Erest Marsde performed a experimet, uder the directio of Erest Rutherford to test Thomso s model of the atom. They used a radioactive source that emitted alpha particles (charge = +) i a tightly collimated beam, oto a gold foil. They the looked at the agle through which the alpha particles are scattered.
Gold Foil Experimet The alpha particle beig relatively (compared to a electro) heavy, should pass through a amorphous positively charged (but dilute) object with little deflectio. What they actually saw was very large deflectios (occasioally) ad relatively large deflectios more ofte. The oly way to explai this was that the positively charged ucleus of the atom was much smaller that the size of the atom! "Geiger-Marsde experimet expectatio ad result" by Kurzo - Ow work. Licesed uder CC BY-SA 3. via Commos - https://commos.wikimedia.org/wiki/file:geiger-marsde_experimet_expectatio_ad_result.svg#/media/file:geiger- Marsde_experimet_expectatio_ad_result.svg
Gold Foil Experimet There are several ways to show the expected results with the Thomso model. The book uses coservatio of eergy. Here, (from Wikipedia) I show usig mometum ad force, that the expected maximum deviatio is quite small. Notice that the agle depeds o the force (~1/r ) ad the time spet uder the force (~r), fially depedig o the radius of the positively charged object as 1/r. QQ r p Ft k r v p QQ 3.41 1.661 k p m rv.36 rad (or.186 ) 19 17 9 8.998 1 7 1 7 6.6451 1.441 (1.531 ) "Thomso model alpha particle scatterig" by Kurzo - Ow work. Licesed uder CC BY-SA 4. via Commos - https://commos.wikimedia.org/wiki/file:thomso_model_alpha_particle_scatterig.svg#/media/file:thomso_model_alpha_part icle_scatterig.svg
Rutherford Model To explai the experimetal results, Rutherford eeded that the radius of positive charge that was resposible for scatterig the alpha particles was roughly 1, times smaller tha the atomic radius. His calculatio of the radius was simple, ad his iterpretatio was correct his alpha particles were t high eough eergy to reach the gold ucleus. 1 1 qq 1 mv 4 b 1 qq 1 14 b.71 m 7 fm 4 mv The actual atomic radius is more like 7.3 fm. He oticed that the deviatios of the 1/r force (from a actual impact) would chage the scatterig agle distributios, which he did t see.
Oops But there was a reaso that Thomso had a plum puddig model. Let s preted agai that we are CLASSICAL physicists. We have a positive ad heavy charge with a small radius, ad electros surroudig it. How do they stay separated from each other? They must be orbitig: 1 qq 1 4 a ma r 1 97e 4 mr But, sice they are acceleratig, they must radiate! This will lose eergy, causig them to fall closer to the positive core I fact, shortly after Thomso's discovery, Hataro Nagaoka, a Japaese physicist, predicted a differet model for electroic structure. Ulike the plum puddig model, the positive charge i Nagaoka's "Saturia Model" was cocetrated ito a cetral core, pullig the electros ito circular orbits remiiscet of Satur's rigs. Few people took otice of Nagaoka's work at the time, ad Nagaoka himself recogized a fudametal defect i the theory eve at its coceptio, amely that a classical charged object caot sustai orbital motio because it is acceleratig ad therefore loses eergy due to electromagetic radiatio. - Wikipedia
Bohr Model I 1913 Rutherford's post-doctoral studet, Niels Bohr, proposed a ew model of the atom. It pre-dated de-broglie waves by about 11 years, so he did t yet kow about matter waves, but he hypothesized that the electro agular mometum was quatized as: L mv r Sice it was quatized, there must be a lowest agular mometum, ad hece, a lowest eergy for the electro. This meat that the electros could t radiate at just ay eergy, ad if the electro was i its lowest eergy, it could t radiate at all. It also could explai emissio ad absorptio spectra, sice electros would oly emit whe chagig eergy from oe quatized state to aother.
Bohr Model Let s get a bit more quatitative. We will examie a electro i the hydroge atom (Z=1), ad start by usig classical forces ad acceleratios: v F ma m r Ad ow solve for the th radius ad the th velocity usig Bohr s quatizatio: L mv r r mv Bohr radius: 1 4 v e r 1 e 1 e mv 4 m r 4 m 1 e v r a 4 me a v m r 4 h 5.91 me 11 m
Bohr Model Ad ow, we ca ask, what is the eergy differece betwee the first two eergy levels? 1 1 e E KE U mv 4 r 1 1 1 e 1 m 4 4 1 1 me 1 4 4 1 m 4 4 4 k me 4 e 4 e 4 me me 13.6 1 ev, where k 4 13.6eV 13.6eV E E 1 1.eV 1 NOTE!!! Differet k!
Hydroge Eergy Levels What frequecy ad wavelegth does this correspod to? E E1 hf 1.eV 1.eV Ufortuately, this is f.4681 15 4.1361 ev s outside of the 8 c 31 m s 11.6m 15 1 visible. f.4681 s But, if we look at the trasitios betwee the third ad secod eergy levels: 13.6eV 13.6eV E3 E 1.89eV 3 15 8 4.1361 ev s31 m s hc 656.5m E 1.89eV 15 Hz
Potificatio o costats Remember the Rydberg costat that was extracted from the hydroge spectra ad Rydberg s reformulatio of the Balmer formula? 4 1 k me 1 E R H k me 1 k e mc 1 k e 1 R mc mc H 4 4 4 c c ke 1 1 here, k c 137 4 Alpha is called the fie structure costat ad it plays a importat role as the couplig costat determiig the stregth of the iteractio betwee electros ad photos!
There is a most profoud ad beautiful questio associated with the observed couplig costat, e the amplitude for a real electro to emit or absorb a real photo. It is a simple umber that has bee experimetally determied to be close to.854455. (My physicist frieds wo't recogize this umber, because they like to remember it as the iverse of its square: about 137.3597 with about a ucertaity of about i the last decimal place. It has bee a mystery ever sice it was discovered more tha fifty years ago, ad all good theoretical physicists put this umber up o their wall ad worry about it.) Immediately you would like to kow where this umber for a couplig comes from: is it related to pi or perhaps to the base of atural logarithms? Nobody kows. It's oe of the greatest dam mysteries of physics: a magic umber that comes to us with o uderstadig by ma. You might say the "had of God" wrote that umber, ad "we do't kow how He pushed his pecil." We kow what kid of a dace to do experimetally to measure this umber very accurately, but we do't kow what kid of dace to do o the computer to make this umber come out, without puttig it i secretly! Richard Feyma, Richard P. Feyma (1985). QED: The Strage Theory of Light ad Matter. Priceto Uiversity Press. p. 19. ISBN -691-8388-6.