PHYS-3301 Lecture 7. CHAPTER 4 Structure of the Atom. Rutherford Scattering. Sep. 18, 2018

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CHAPTER 4 Structure of the Atom PHYS-3301 Lecture 7 4.1 The Atomic Models of Thomso ad Rutherford 4.2 Rutherford Scatterig 4.3 The Classic Atomic Model 4.4 The Bohr Model of the Hydroge Atom 4.

1 CHAPTER 4 Structure of the Atom PHYS-3301 Lecture The Atomic Models of Thomso ad Rutherford 4.2 Rutherford Scatterig 4.3 The Classic Atomic Model 4.4 The Bohr Model of the Hydroge Atom 4.5 Successes ad Failures of the Bohr Model 4.6 Characteristic X-Ray Spectra ad Atomic Number 4.7 Atomic Excitatio by Electros Sep. 18, 2018 Rutherford Scatterig Scatterig experimets help us study matter too small to be observed directly. There is a relatioship betwee the impact parameter b ad the scatterig agle θ. The Relatioship Betwee the Impact Parameter b ad the Scatterig Agle q Whe b is small, r gets small. Coulomb force gets large. θ ca be large ad the particle ca be repelled backward. Figure 4.7 The relatioship betwee the impact parameter b ad scatterig agle q. Particles with small impact parameters approach the ucleus most closely (r mi ) ad scatter to the largest agles. Particles withi the rage of impact parameters b will be scattered withi Dq

2 Rutherford Scatterig Ay particle iside the circle of area πb 02 will be similarly scattered. Rutherford Scatterig Equatio I actual experimet a detector is positioed from θ to θ + dθ that correspods to icidet particles betwee b ad b + db. If we have a target foil of thickess t with atoms/volume, the # of target uclei per uit area = t The cross sectio σ = πb 2 is related to the probability for a particle beig scattered by a ucleus. The fractio of icidet particles scattered is = tas / A = ts = t(pb 2 ) The umber of particles scattered per uit area is The umber of scatterig uclei per uit area. N(q) = N i df / da N i = total umber of icidet particles The Importat Poits 1. The scatterig is proportioal to the square of the atomic umber of both the icidet particle (Z 1 ) ad the target scatterer (Z 2 ). 2. The umber of scattered particles is iversely proportioal to the square of the kietic eergy of the icidet particle. 3. For the scatterig agle, the scatterig is proportioal to 4 th power of si( /2). 4. The Scatterig is proportioal to the target thickess for thi targets. 4.3: The Classical Atomic Model As suggested by the Rutherford Model the atom cosisted of a small, massive, positively charged ucleus surrouded by movig electros. This the suggested cosideratio of a plaetary model of the atom. Let s cosider atoms as a plaetary model.

3 4.3: The Classical Atomic Model As suggested by the Rutherford Model the atom cosisted of a small, massive, positively charged ucleus surrouded by movig electros. This the suggested cosideratio of a plaetary model of the atom. Let s cosider atoms as a plaetary model. The force of attractio o the electro by the ucleus ad Newto s 2d law give The Plaetary Model is Doomed From classical E&M theory, a accelerated electric charge radiates eergy (electromagetic radiatio) which meas total eergy must decrease. Radius r must decrease!! where v is the tagetial velocity of the electro. Electro crashes ito the ucleus!? The total eergy is Physics had reached a turig poit i 1900 with Plack s hypothesis of the quatum behavior of radiatio. 4.4: The Bohr Model of the Hydroge Atom Bohr s dramatic geeral assumptios: A. Statioary states or orbits must exist i atoms, i.e., orbitig electros do ot radiate eergy i these orbits. These orbits or statioary states are of a fixed defiite eergy E. B. The emissio or absorptio of electromagetic radiatio ca occur oly i cojuctio with a trasitio betwee two statioary states. The frequecy, f, of this radiatio is proportioal to the differece i eergy of the two statioary states: E = E 1 E 2 = hf where h is Plack s Costat C. Classical laws of physics do ot apply to trasitios betwee statioary states. D. The mea kietic eergy of the electro-ucleus system is K = hf orb /2, where f orb is the frequecy of rotatio. This is equivalet to the agular mometum of a statioary state to be a itegral multiple of h/2 (= h bar) Bohr Radius

4 Bohr Radius The Hydroge Atom The eergies of the statioary states The diameter of the hydroge atom for statioary states is Where the Bohr radius is give by The smallest diameter of the hydroge atom is = 1 gives its lowest eergy state (called the groud state) The Hydroge Atom The eergies of the statioary states Trasitios i the Hydroge Atom Lyma series where E 0 = 13.6 ev Emissio of light occurs whe the atom is i a excited state ad decays to a lower eergy state ( u l ). The atom will remai i the excited state for a short time before emittig a photo ad returig to a lower statioary state. All hydroge atoms exist i = 1 (ivisible). where f is the frequecy of a photo. Balmer series R is the Rydberg costat. Whe sulight passes through the atmosphere, hydroge atoms i water vapor absorb the wavelegths (visible).

5 Fie Structure Costat Fie Structure Costat The electro s velocity i the Bohr model: O the groud state, v 1 = m/s ~ less tha 1% of the speed of light The ratio of v 1 to c is the fie structure costat. The Correspodece Priciple The Correspodece Priciple The frequecy of the radiatio emitted f classical is equal to the orbital frequecy f orb of the electro aroud the ucleus. Classical electrodyamics + Bohr s atomic model The frequecy of the trasitio from + 1 to is Determie the properties of radiatio Need a priciple to relate the ew moder results with classical oes. Bohr s correspodece priciple I the limits where classical ad quatum theories should agree, the quatum theory must reduce the classical result. For large, Substitute E 0 : So the frequecies of the radiated eergy betwee classical theory & the Bohr model for large values of the quatum #.

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