Harris Chapter 7 - Atomic Structure

1. Harris Chapter 7- Atomic Structure • 7.1 • Orbital Magnetic Moments, discovery of intrinsic spin • 7.2 & 7.3 • Identical Particles (warning: examples in book all inf-squ well) • Exclusion Principle • 7.4 & 7.5 • Multielectron Atoms, effective charges • Hartree Treatment • 7.6 • Spin-Orbit Effect • 7.7 • Adding QM Angular Momenta • 7.9 & 7.8 • Multielectron Spectroscopic Notation • Zeeman Effect

2. http://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgihttp://asd-www.larc.nasa.gov/cgi-bin/SCOOL_Clouds/Cumulus/list.cgi Summary So Far

3. 7.1 Orbital Magnetic Moments and Discovery of Intrinsic Spin

4. Two kinds of Angular Momentum • Classical Angular Momentum • L = r x p • r vector, p vector  L vector • L obeys vector math • Any L possible, no contraints on Lx Ly Lz • Quantum • Quantum Mechanical Angular Momentum • L = r x p • r vector, p vector operator  L 3 component operator • L obeys …… got to be careful • L described by two labels l , m • L and Lz can be known, Lx and Ly cannot

5. Bohr Model of Ang Momentum Classical or Semi-classical description Note: s-states (l=0) have no Bohr model picture Eisberg & Resnick: Fig 7-11

6. Vector Model of QM Ang. Momentum quantum numbers E&R Fig 7-12

7. Edmonds“A.M. in QM” pg 19: “We might imagine the vector moving in an unobservable way about the z-axis...” pg 29: “The QM probability density, not being time dependent, gives us no information about the motion of the particle in it’s orbit.” Y*(r,t) Y(r,t) Y(r,t)=Y(r) e-iwt

8. Morrison, Estle, Lane “Understanding More QM”, Prentice-Hall, 1991

9. Otto Stern & Walther Gerlach~1922 Bohr’s Q hypothesis Sommerfeld’s Q hypothesis 3 2 1 Assigned by advisor Max Born to demonstrate existence of the l, ml quantum numbers

10. Orbital Magnetic Moment E&R Fig 7-11

11. E&R Fig 7-11 Orbital Magnetic Moment

12. E&R Fig 7-11 Bohr magneton

13. B m

14. B m Different ml states experience different forces

15. Use B as z-axis. Different ml states experience different forces

16. Stern & Gerlach~1922 Harris Fig 7.3, 7.4

17. Stern & Gerlach~1922 Intended to demonstrate space quantization (l), & therefore expected odd number of spots, but observed an even number. http://upload.wikimedia.org/wikipedia/en/2/29/Stern-Gerlach_experiment.PNG

18. Despite Stern's careful design and feasibility calculations, the experiment took more than a year to accomplish. In the final form of the apparatus, a beam of silver atoms (produced by effusion of metallic vapor from an oven heated to 1000°C) was collimated by two narrow slits (0.03 mm wide) and traversed a deflecting magnet 3.5 cm long with field strength about 0.1 tesla and gradient 10 tesla/cm. The splitting of the silver beam achieved was only 0.2 mm. Accordingly, misalignments of collimating slits or the magnet by more than 0.01 mm were enough to spoil an experimental run. The attainable operating time was usually only a few hours between breakdowns of the apparatus. Thus, only a meager film of silver atoms, too thin to be visible to an unaided eye, was deposited on the collector plate. Stern described an early episode: http://www.physicstoday.org/pt/vol-56/iss-12/p53.html

19. Stern described an early episode: After venting to release the vacuum, Gerlach removed the detector flange. But he could see no trace of the silver atom beam and handed the flange to me. With Gerlach looking over my shoulder as I peered closely at the plate, we were surprised to see gradually emerge the trace of the beam. . . . Finally we realized what [had happened]. I was then the equivalent of an assistant professor. My salary was too low to afford good cigars, so I smoked bad cigars. These had a lot of sulfur in them, so my breath on the plate turned the silver into silver sulfide, which is jet black, so easily visible. It was like developing a photographic film.7 http://www.physicstoday.org/pt/vol-56/iss-12/p53.html

20. Wolfgang Pauli ~ 1924 • Pauli Exclusion Principle • No two electrons can have the same quantum number • Postulated an additional quantum number (i.e. label) • Believed it came from the interaction between electrons.

21. Ralph Kronig ~1925 • Spinning Electron Idea

22. Goudsmit & Ulhenbeck ~ 1925 • Studied high resolution spectra of alkali elements

23. Ocean Optics - Helium

24. Ocean Optics - Neon

25. Giancoli – fig 36.21

26. The old and the new term scheme of hydrogen [5]. The scheme shows the multiplet splitting of the excited states of the hydrogen atom with principal quantum number n=3, presented by Goudsmit in the form in which it appeared in the original publications of1926. The assignment in the current notation has been added at the right. With the development of quantum mechanics the notation changed. The quantum numbers L and J now usedfor the orbital and total angular momentum, respectively, correspond to K-1/2 and J-1/2 in the figure. The "forbidden component" referred to by Goudsmit is of the type 3 2P1/2--> 2 2S in which the total angular momentum is conserved and L changes by plus or minus 1. [5] S. Goudsmit and G.E. Uhlenbeck, Physica 6 (1926) 273.

27. Uhlenbeck & Goudsmit~ 1925 The discovery note in Naturwissenschaften is dated 17 October 1925. One day earlier Ehrenfest had written to Lorentz to make an appointment and discuss a "very witty idea" of two of his graduate students. When Lorentz pointed out that the idea of a spinning electron would be incompatible with classical electrodynamics, Uhlenbeck asked Ehrenfest not to submit the paper. Ehrenfest replied that he had already sent off their note, and he added: "You are both young enough to be able to afford a stupidity!" http://www.lorentz.leidenuniv.nl/history/spin/spin.html

28. Uhlenbeck & Goudsmit~ 1925 Ehrenfest's encouraging response to his students ideas contrasted sharply with that of Wolfgang Pauli. As it turned out, Ralph Kronig, a young Columbia University PhD who had spent two years studying in Europe, had come up with the idea of electron spin several months before Uhlenbeck and Goudsmit. He had put it before Pauli for his reactions, who had ridiculed it, saying that "it is indeed very clever but of course has nothing to do with reality". Kronig did not publish his ideas on spin. No wonder that Uhlenbeck would later refer to the "luck and privilege to be students of Paul Ehrenfest". http://www.lorentz.leidenuniv.nl/history/spin/spin.html

29. “This isn't right. This isn't even wrong.” http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html There were some people thinking about electron spin in those days, but there was a lot of basic opposition to such an idea. One of the first was Ralph de Laer Kronig. He got the idea that the electron should have a spin in addition to its orbital motion. He was working with Wolfgang Pauli at the time, and he told his idea to Pauli. Pauli said, "No, it's quite impossible." Pauli completely crushed Kronig. Then the idea occurred quite independently to two Young Dutch physicists, George Uhlenbeck and Samuel Goudsmit. They were working in Leiden with Professor Paul Ehrenfest, and they wrote up a little paper about it and took it to Ehrenfest. Ehrenfest liked the idea very much. He suggested to Uhlenbeck and Goudsmit that they should go and talk it over with Hendrik Lorentz, who lived close by in Haarlem. His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect" (Frisch 1991, p. 48; Gamow 1985). "The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

30. “This isn't right. This isn't even wrong.” http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html They did go and talk it over with Lorentz. Lorentz said, "No, it's quite impossible for the electron to have a spin. I have thought of that myself, and if the electron did have a spin, the speed of the surface of the electron would be greater than the velocity of light. So, it's quite impossible." Uhlenbeck and Goudsmit went back to Ehrenfest and said they would like to withdraw the paper that they had given to him. Ehrenfest said, "No, it's too late; I have already sent it in for publication " His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect" (Frisch 1991, p. 48; Gamow 1985). "The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

31. The calculation(using current values) r < 2.8 E-19 m b > 3 * 10 + 6 value from Bhabha scattering at CERN

32. “This isn't right. This isn't even wrong.” http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Pauli.html That is how the idea of electron spin got publicized to the world. We really owe it to Ehrenfest's impetuosity and to his not allowing the younger people to be put off by the older ones. The idea of the electron having two states of spin provided a perfect answer to the duplexity. His ability to make experiments self destruct simply by being in the same room was legendary, and has been dubbed the "Pauli effect" (Frisch 1991, p. 48; Gamow 1985). "The Birth of Particle Physics," edited by Laurie M. Brown and Lillian Hoddeson. The essay by Paul A.M. Dirac is entitled "Origin of Quantum Field Theory."

33. Letter fm Thomas to Goudsmit Part of a letter by L.H. Thomas to Goudsmit (25 March 1926). Reproduced from a transparency shown by Goudsmit during his 1971 lecture. The original is presumably in the Goudsmit archive kept by the AIP Center for History of Physics. http://www.lorentz.leidenuniv.nl/history/spin/goudsmit.html

34. intrinsic spin • Fundamental objects • electron spin – ½ • neutrino spin – ½ , but LH only • photon spin – 1 • Composite objects • proton spin – ½ • neutron spin – ½ • D delta spin – 3/2

35. How to Denote Wavefunctions(version 1) the spinor has no ‘functional form’ because spin is not a spatial feature

36. Two types of Magnetic Moments L S

37. interesting fundamental constants -2.002 319 304 3622 (15) 1.602 176 487 (40) x 10-19 C

38. 7.2 & 7.3 Complications from having Identical Particles

39. Exchange Symmetry

40. 7.4 & 7.5 Multielectron Atoms

41. En = ( -13.6 eV ) (Z2/n2) r = n2 ao / Z ao = 0.529 Å

42. Prob = r2 R* R

43. orbitals get sucked down the most Crossings occur for the upper orbitals 4p 3d 4s 3p 3s 2p 2s 1s 1s sucked off bottom of page

44. Note: This shows how the orbitals shift as viewed from the perspective of an s-orbital.

45. Hartree-Fock Method

46. Hartree-Fock Methods Choose initial shape For Coulomb Potl V(r) Solve Schro Eqn for EnYn Insert fine structure corrections Build atom according to This set of orbital energies En Loop until V(r) doesn’t change much Use the collection of Yn*Yn to Get new electron charge distrib Use Gauss’ Law to get new V(r) shape

48. Using effective charge is a very crude approximation.

49. r2 ~ n2 ao / Zeff Hartree-FockEffective Charge Effects En ~ (Zeff2/n2) ( -13.6 eV )