CHAPTER 4 Structure of the Atom Fall 2018 Prof. Sergio B. Mendes 1
Topics 4.1 The Atomic Models of Thomson and Rutherford 4.2 Rutherford Scattering 4.3 The Classic Atomic Model 4.4 The Bohr Model of the Hydrogen Atom 4.5 Successes and Failures of the Bohr Model 4.6 Characteristic X-Ray Spectra and Atomic Number 4.7 Atomic Excitation by Electrons Fall 2018 Prof. Sergio B. Mendes 2
4.1 The Atom by Early 1900 s Atom size 10 10 mm = 1 A = 0.1 nnnn Atom mass = mm aaaaaaaa mm ee Atom charge = qq aaaaaaaa 0, neutral Fall 2018 Prof. Sergio B. Mendes 3
J.J. Thomson Plum Pudding Model of the Atom: electrons distributed over a fluid-like volume containing most of the mass and the positive charges atom electron Fall 2018 Prof. Sergio B. Mendes 4
PhET on Scattering with Thomson Atom Fall 2018 Prof. Sergio B. Mendes 5
4.2 Rutherford Scattering, 1911 αα-particles: doubly ionized He-atom Some αα-particles undergo large scattering angles Large scattering angles are not expected with Thomson s model Nuclear model: positive charges and mass are concentrated in a small fraction of the whole atom Fall 2018 Prof. Sergio B. Mendes 6
Talk Softly Please Fall 2018 Prof. Sergio B. Mendes 7
PhET on Rutherford Scattering Fall 2018 Prof. Sergio B. Mendes 8
How to describe the problem: Representation of the Rutherford scattering: the αα-particle of mass mm and charge ZZ 11 ee scatters from a particle of charge ZZ 22 ee at rest. The parameters rr and φφ, which describe the projectile s orbit, are defined as shown. The angle φφ = 00 corresponds to the position of closest approach. The impact parameter bb and scattering angle θθ are also displayed. Fall 2018 Prof. Sergio B. Mendes 9
Assumptions of the Model The scatterer is so massive that it does not significantly recoil, its kinetic energy does not change. (Later the calculations were extended to include recoil) The target is thin such that only one scattering event occur for each αα-particle. The αα-particle and the target scatterer are small and considered as point masses and charges. The energies are low enough that Coulomb force is sufficient to describe the collision event (nuclear forces can be neglected). Fall 2018 Prof. Sergio B. Mendes 10
Angular Momentum is Constant!! FF = 1 ZZ 1 ZZ 2 ee 2 4 ππ εε oo rr 3 rr pp rr ττ = rr FF = ddll dddd ττ = 0 = ddll dddd LL = rr pp = cccccccccccccccc Scattered particle stays in a plane!! LL = bb mm vv oo Fall 2018 Prof. Sergio B. Mendes 11
Change in Linear Momentum FF = pp tt pp = FF dddd Fall 2018 Prof. Sergio B. Mendes 12
Bisector of the Isosceles Triangle pp = FF dddd pp = FF pp dddd pp 2 = mm vv oo ssssss θθ 2 Fall 2018 Prof. Sergio B. Mendes 13
The Change in Linear Momentum: pp = FF pp dddd = FF cccccc φφ dddd LL = rr pp = bb mm vv oo = rr mm rr dddd dddd dddd rr 2 = dddd bb vv oo = = ZZ 1 ZZ 2 ee 2 1 cccccc φφ dddd 4 ππ εε oo rr2 ZZ 1 ZZ 2 ee 2 φφ ff cccccc φφ dddd 4 ππ εε oo bb vv oo Fall 2018 Prof. Sergio B. Mendes 14 φφ ii
φφ ff cccccc φφ dddd = ssssss φφ ff ssssss φφ ii φφ ii = 2 cccccc θ 2 φφ ii + φφ ff = π θ φφ ii = φφ ff= π θ 2 pp = ZZ 1 ZZ 2 ee 2 φφ ff cccccc φφ dddd 4 ππ εε oo bb vv oo φφ ii = ZZ 1 ZZ 2 ee 2 4 ππ εε oo bb vv oo 2 cccccc θ 2 Fall 2018 Prof. Sergio B. Mendes 15
Relation of Impact Parameter and Angle of Deflection 2 mm vv oo ssssss θθ 2 = = ZZ 1 ZZ 2 ee pp 2 2 cccccc θ 4 ππ εε oo bb vv oo 2 bb = ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv 2 cccctt θ oo 2 Fall 2018 Prof. Sergio B. Mendes 16
Fall 2018 Prof. Sergio B. Mendes 17
Fall 2018 Prof. Sergio B. Mendes 18
Number of nucleus per unit area nn = # oooo nnnnnnnnnnnnnn vvvvvvvvvvvv = # oooo nnnnnnnnnnnnnn aaaaaaaa ttttttttttttttttt nn tt = # oooo nnnnnnnnnnnnnn aaaaaaaa Fall 2018 Prof. Sergio B. Mendes 19
Probability of an α-particle between bb and bb + dddd of a nucleus bb ddbb nn tt aaaaaaaa = nn tt 2 π bb dddd = PP bb dddd = pp θθ ddθθ Fall 2018 Prof. Sergio B. Mendes 20
pp θθ = nn tt 2 π bb dddd dddd bb = ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv oo 2 cccctt θ 2 pp θθ = nn tt 2 π ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv 2 oo 2 ssssss θθ 4 ssssss 4 θθ 2 Fall 2018 Prof. Sergio B. Mendes 21
Rutherford Scattering Equation pp θθ = nn tt ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv oo 2 2 2 π ssssss θθ 4 ssssss 4 θθ 2 pp θθ ddθθ = pp Ω ddω = pp Ω 2 π ssssss θθ ddθθ pp ΩΩ = nn tt ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv oo 2 2 1 4 ssssss 4 θθ 2 Fall 2018 Prof. Sergio B. Mendes 22
Experimental Confirmation Results of undergraduate laboratory experiment of scattering 1-MeV protons from a gold target. The solid line shows the ssssss 4 θθ 2 angular dependence of the data, verifying Rutherford s calculation. Fall 2018 Prof. Sergio B. Mendes 23
Highlights of Rutherford s Results Dependence on ssssss 4 θθ 2 magnitude confirmed over 5 orders of Dependence on tt confirmed over a factor of 10 Confirmed with several kinetic energy from different radioactive sources Allows determination of Z for several atoms Fall 2018 Prof. Sergio B. Mendes 24
Distance of Closest Approach: RR RR = ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv oo 2 1 + 1 ssssss θθ 2 Smallest R bb = 0 θθ = ππ RR cccccccccccccc = 2 ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv oo 2 Fall 2018 Prof. Sergio B. Mendes 25
Determining the Nucleus Radius: pp θθ = nn tt 2 π ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv oo 2 2 ssssss θθ 4 ssssss 4 θθ 2 = nn tt π 8 RR cccccccccccccc 2 ssssss θθ ssssss 4 θθ 2 RR cccccccccccccc 2 = ZZ 1 ZZ 2 ee 2 4 ππ εε oo mm vv oo 2 Nuclear radius 10 14 mm Atom radius 10 10 mm Fall 2018 Prof. Sergio B. Mendes 26
4.3 The Classical (and incorrect) Atomic Model (atom with 1 electron: H, He +, Li ++, ) mm ee ee rr ZZee FF = ZZ ee 2 4 ππ εε oo rr 2 = mm ee aa = mm ee vv 2 aa = vv2 rr rr vv 2 = ZZ ee 2 4 ππ εε oo mm ee rr Fall 2018 Prof. Sergio B. Mendes 27
Kinetic and Potential Energy EE = KK + UU = 1 2 mm ee vv 2 ZZ ee 2 4 ππ εε oo rr vv 2 = ZZ ee 2 4 ππ εε oo mm ee rr EE = ZZ ee 2 8 ππ εε oo rr Fall 2018 Prof. Sergio B. Mendes 28
The Classical Atom is not stable!! An electric charge under acceleration radiates energy (source of electromagnetic radiation)!! Smaller total energy means smaller radius Classical calculations using electrodynamic theory predicts that the charges will spiral towards the nucleus in a fraction of a second (picosecond) Fall 2018 Prof. Sergio B. Mendes 29
4.4 The Bohr Model of the one electron atom 30
Bohr s Postulates: 1. An electron in an atom moves in a circular orbit about the nucleus under the influence of the Coulomb attraction between the electron and the nucleus, obeying the laws of classical mechanics. 2. Instead of the infinite number of orbits which would be possible in classical mechanics, it is only possible for an electron to move in an orbit for which its orbital angular momentum, LL, is an integral multiple of h/2ππ LL = nn h 2ππ = nn ħ Fall 2018 Prof. Sergio B. Mendes 31
Additional Postulates: 3. Despite the fact that it is constantly accelerating, an electron moving in such allowed orbit does not radiate electromagnetic energy. Thus, its total energy E remains constant, as long as it stays in such allowed orbit. 4. Electromagnetic radiation is emitted if an electron, initially moving one allowed orbit of total energy EE ii, discontinuously changes its motion so that it moves to another allowed orbit of total energy EE ff. The frequency of the emitted (or absorbed) radiation is given by. ff = EE ii EE ff h Fall 2018 Prof. Sergio B. Mendes 32
From the Coulomb force for a circular orbit: FF = ZZ ee 2 4 ππ εε oo rr 2 = mm ee vv 2 rr vv = ZZ ee 2 4 ππ εε oo mm ee rr 1/2 Fall 2018 Prof. Sergio B. Mendes 33
Angular Momentum: vv = ZZ ee 2 4 ππ εε oo mm ee rr 1/2 LL = rr mm ee vv ZZ ee 2 = rr mm ee 4 ππ εε oo mm ee rr 1/2 Fall 2018 Prof. Sergio B. Mendes 34
From Postulate (2): LL = rr mm ZZ ee 2 4 ππ εε oo mm ee rr 1/2 = nn ħ rr nn = nn 2 1 ZZ 4 ππ εε oo ħ 2 mm ee ee 2 = nn 2 1 ZZ aa oo Bohr radius aa oo 4 ππ εε oo ħ 2 mm ee ee 2 = 0.529 10 10 mm Fall 2018 Prof. Sergio B. Mendes 35
Hydrogen: Z = 1 rr nn = nn 2 1 ZZ aa oo rr nn = nn 2 aa oo Smallest radius: n = 1 rr 1 = aa oo = 0.529 10 10 mm Fall 2018 Prof. Sergio B. Mendes 36
Kinetic and Potential Energy EE = KK + UU = 1 2 mm ee vv 2 ZZ ee 2 4 ππ εε oo rr vv 2 = ZZ ee 2 4 ππ εε oo mm ee rr EE = ZZ ee 2 8 ππ εε oo rr Fall 2018 Prof. Sergio B. Mendes 37
Total Energy EE = ZZ ee 2 8 ππ εε oo rr rr nn = nn 2 1 ZZ 4 ππ εε oo ħ 2 mm ee ee 2 EE nn = ZZ ee 2 8 ππ εε oo rr nn = ZZ2 nn 2 mm ee ee 4 32 ππ 2 εε oo 2 ħ 2 = ZZ2 nn 2 EE oo EE oo mm ee ee 4 32 ππ 2 εε oo 2 ħ 2 = 13.6 eeee Fall 2018 Prof. Sergio B. Mendes 38
Hydrogen: Z = 1 EE nn = 1 13.6 eeee nn2 Fall 2018 Prof. Sergio B. Mendes 39
From Postulate (4): ff = cc λλ ff = EE ii EE ff h h ff = EE ii EE ff EE nn = ZZ2 nn 2 EE oo h cc λλ = 1 nn ii 2 1 nn ff 2 ZZ2 EE oo 1 λλ = 1 nn 2 1 ff nn 2 ii ZZ2 EE oo hcc Fall 2018 Prof. Sergio B. Mendes 40
Hydrogen: Z = 1 1 λλ = 1 nn 2 1 ff nn 2 ii EE oo h cc EE oo mm ee ee 4 32 ππ 2 εε oo 2 ħ 2 RR EE oo h cc Fall 2018 Prof. Sergio B. Mendes 41
Rydberg Equation, 1890 1 λλ = RR HH 1 nn 2 1 kk 2 kk > nn Fall 2018 Prof. Sergio B. Mendes 42
PhET Fall 2018 Prof. Sergio B. Mendes 43
4.5 Successes of Bohr s Model Fall 2018 Prof. Sergio B. Mendes 44
Reduced Mass Correction: 1 1 + 1 μμ ee mm ee MM ZZee EE oo, EE oo mm ee ee 4 32 ππ 2 εε oo 2 ħ 2 = h cc RR μμ ee ee 4 32 ππ 2 εε oo 2 ħ 2 = h cc RR RR RR = μμ ee mm ee = RR = RR 1 1 1 + mm ee MM ZZZZ 1 + mm ee MM ZZZZ Fall 2018 Prof. Sergio B. Mendes 45
Hydrogen, Deuterium, Tritium MM PPPPPPPPPPPP = 1.007276 u 1 pppppppppppp MM DDDDDDDDDDDDDDDD = 2.013553 u 1 pppppppppppp + 1 nnnnnnnnnnnnnn MM TTTTTTTTTTTT = 3.015500 u 1 pppppppppppp + 2 nnnnnnnnnnnnnnnn ZZ = 1 1 RR HH = RR 1 + mm ee MM PPPPPPPPPPPP 1 RR DD = RR mm 1 + ee MM DDDDDDDDDDDDDDDD RR TT = RR 1 1 + mm ee MM TTTTTTTTTTTT Fall 2018 Prof. Sergio B. Mendes 46
Changes in Observed Spectral Lines RR HH = RR 1 1 1 + mm ee MM PPPPPPPPPPPP λλ = RR 1 2 2 1 3 2 λλ = 656.47 nnnn 1 RR DD = RR mm 1 + ee MM DDDDDDDDDDDDDDDD RR TT = RR 1 1 + mm ee MM TTTTTTTTTTTT λλ = 656.29 nnnn λλ = 656.23 nnnn Fall 2018 Prof. Sergio B. Mendes 47
One-Electron Atoms: H, He +, Li ++ ZZ = 1, 2, 3 1 λλ = 1 nn 2 1 ff nn 2 ii ZZ2 EE oo hcc EE oo μμ ee ee 4 32 ππ 2 εε oo 2 ħ 2 H: 1 λλ = 1 nn 2 1 ff 2 2 12 EE oo hcc He + : 1 λλ = 1 nn 2 1 ff 4 2 22 EE oo hcc Fall 2018 Prof. Sergio B. Mendes 48
4.6 X Ray Emission Spectra Fall 2018 Prof. Sergio B. Mendes 49
EE ff + h cc λλ = EE ii = ee VV oo λλ = h cc ee VV oo EE ff The relative intensity of x rays produced in an x- ray tube is shown for an accelerating voltage of 35 kv. Fall 2018 Prof. Sergio B. Mendes 50
Henry Moseley Henry G. J. Moseley (1887 1915), shown here working in 1910 in the Balliol- Trinity laboratory of Oxford University, was a brilliant young experimental physicist with varied interests. Unfortunately, he was killed in action at the young age of 27 during the English expedition to the Dardanelles. Moseley volunteered and insisted on combat duty in World War I, despite the attempts of Rutherford and others to keep him out of action Fall 2018 Prof. Sergio B. Mendes 51
Moseley Data, 1914 ff = 3 cc RR 4 ZZ 1 2 Fall 2018 Prof. Sergio B. Mendes 52
Extending Bohr s model to multi-electron atoms: ff = 3 cc RR 4 ZZ 1 2 1 λλ KKαα = ZZ 1 2 RR 1 1 2 1 2 2 = ZZ 1 2 RR 3 4 1 = ZZ 1 2 RR λλ KK 1 1 nn 2 1 = ZZ 2 λλ eeeeee RR LL 1 2 2 1 nn 2 Fall 2018 Prof. Sergio B. Mendes 53
4.7 Atomic Excitation by Electrons Fall 2018 Prof. Sergio B. Mendes 54
Frank-Hertz Experiment Fall 2018 Prof. Sergio B. Mendes 55
Electron Excitation of Hg Atoms Fall 2018 Prof. Sergio B. Mendes 56
Current versus Voltage Data: Fall 2018 Prof. Sergio B. Mendes 57
Interpretation of Data: Fall 2018 Prof. Sergio B. Mendes 58
A Few Remarks Fall 2018 Prof. Sergio B. Mendes 59
The Correspondence Principle Or, what happen for large n? ff = 1 nn ff 2 1 nn ii 2 EE oo h nn ii = nn + 1 nn = nn ff ff 2 EE oo h nn 3 mm ee ee 4 EE oo 32 ππ 2 εε 2 oo ħ 2 ff cccc = 1 2 ππ ωω = 1 2 ππ ff ff cccc vv rr vv nn = 1 ee 2 4 ππ εε oo mm ee rr nn rr nn = nn 2 4 ππ εε oo ħ 2 mm ee ee 2 1/2 Fall 2018 Prof. Sergio B. Mendes 60
Difficulties with the Bohr s Model It applied only to single-electron atoms. It couldn t explain the intensities of the spectral lines. It wasn t able to explain the fine structure of several spectral lines. It didn t provide an explanation for the binding of atoms into molecules. Conceptually the theory was not consistent. At certain aspects, classical mechanics was considered valid, but for certain points it was replaced with new postulates. Fall 2018 Prof. Sergio B. Mendes 61
Appendix A: Reduced Mass rr CCCC mm 1 rr 1 + mm 2 rr 2 mm 1 + mm 2 rr rr 2 rr 1 rr 1 = rr 2 rr 1 rr 1 = rr CCCC + rr 1 rr 1 rr CCCC rr 2 rr 2 = rr CCCC + rr 2 rr 2 mm 1 rr 1 = mm 1 rr CCCC + mm 1 rr 1 mm 2 rr 2 = mm 2 rr CCCC + mm 2 rr 2 0 = mm 1 rr 1 + mm 2 rr 2 Fall 2018 Prof. Sergio B. Mendes 62
mm 1 rr 1 = mm 2 rr 2 rr rr 2 rr 1 = rr 2 rr 1 rr 1 = mm 2 mm 1 + mm 2 rr rr 2 = + mm 1 mm 1 + mm 2 rr Fall 2018 Prof. Sergio B. Mendes 63
rr 1 = mm 2 mm 1 + mm 2 rr rr 1 = rr CCCC + rr 1 = rr CCCC mm 2 mm 1 + mm 2 rr rr 2 = + mm 1 mm 1 + mm 2 rr rr 2 = rr CCCC + rr 2 = rr CCCC + mm 1 mm 1 + mm 2 rr Fall 2018 Prof. Sergio B. Mendes 64
rr 1 = rr CCCC mm 2 mm 1 + mm 2 rr vv 1 = vv CCCC mm 2 mm 1 + mm 2 vv rr 2 = rr CCCC + mm 1 mm 1 + mm 2 rr vv 2 = vv CCCC + mm 1 mm 1 + mm 2 vv 1 2 mm 1 vv 2 1 + 1 2 mm 2 vv 2 2 = 1 2 mm 1 + mm 2 vv 2 CCCC + 1 2 mm 1 mm 2 mm 1 + mm 2 vv 2 μμ mm 1 mm 2 mm 1 + mm 2 Fall 2018 Prof. Sergio B. Mendes 65