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My ABC System of Spectroscopy

My ABC System of Spectroscopy. Rotational Spectroscopy. A Classical Description > E = T + V E = ½I  2 V=0 B QM description > the Hamiltonian H  J  = E  J  H = J 2 /2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1)

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My ABC System of Spectroscopy

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  1. My ABC System of Spectroscopy

  2. Rotational Spectroscopy

  3. A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J =±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc

  4. Rotational Spectra Linear Molecules E = ½I2

  5. Rigid Diatomic molecule Rotational Spectra Linear Molecules E = ½I2

  6. Rigid Diatomic molecule Angular velocity  Rotational Spectra Linear Molecules E = ½I2

  7. Rigid Diatomic molecule Angular velocity  Rotational Spectra Linear Molecules E = ½I2 m2 m1

  8. Rigid Diatomic molecule Angular velocity  Rotational Spectra Linear Molecules E = ½I2 m2 m1 I = r2

  9. Rigid Diatomic molecule Angular velocity  Rotational Spectra Linear Molecules E = ½I2 m2 m1 • I = r2 • = m1m2/(m1+m2)

  10. m = 16 For carbon monoxide CO m = 12 • = m1m2/(m1+m2) = 12x16/(12+16) = 12x16/28

  11. A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J =±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc

  12. Rotational Spectra Linear Molecules E = ½I2  J2/2I (J = I )

  13. Rotational Spectra Linear Molecules E = ½I2  J2/2I (J = I ) E = ½ mv2  p2/2m (p = mv)

  14. Rotational Spectra Linear Molecules E = ½I2  J2/2I (J = I ) E = ½ mv2  p2/2m (p = mv) H = J2/2I (Note V= 0)

  15. A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J =±1 E Transition Frequencies > F B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc

  16. H = J2/2I

  17. H = J2/2I J J2J  =ħ2 J(J+1)

  18. H = J2/2I J J2J  =ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1)

  19. H = J2/2I J J2J  =ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1) F(J) = B J(J+1)

  20. H = J2/2I J J2J  =ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1) F(J) = B J(J+1) B = ħ2/h2I MHz B = ħ2/hc2I cm-1

  21. H = J2/2I J J2J  =ħ2 J(J+1) E(J) = (ħ2/2I) J(J+1) F(J) = B J(J+1) B = ħ2/h2I MHz B = ħ2/hc2I cm-1 J J2J  J*J2 Jd

  22. Rigid Diatomic molecule Angular velocity  Rotational Spectra Linear Molecules E = ½I2 m2 m1 • I = r2 • = m1m2/(m1+m2) B (MHz) = 505391/I (uÅ 2) B (cm-1) = 16.863/I (uA2)

  23. Take a sheet of lined paper and assign the line spacing as 2B

  24. Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 0

  25. Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 1 2B 0

  26. Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 2 6B 1 2B 0

  27. Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 3 12B 2 6B 1 2B 0

  28. Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 4 20B 3 12B 2 6B 1 2B 0

  29. Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… 5 30B 4 20B 3 12B 2 6B 1 2B 0

  30. Rotational Spectroscopy of Linear Molecules F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  31. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 0 2B 6B 12B 20B 30B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  32. A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J =±1 E Transition Frequencies > F = 2B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc

  33. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  34. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  35. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  36. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  37. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  38. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  39. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  40. Rotational Spectroscopy of Linear Molecules J 7 56B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0

  41. Rotational Spectroscopy of Linear Molecules J 7 56B 14B F(J) = BJ(J+1) 2B 6B 12B 20B 30B… F(J) = 2B(J+1) 2B 4B 6B 8B 10B 12B… 6 42B 12B 5 30B 10B 4 20B 8B 3 12B 6B 2 6B 4B 1 2B 2B 0

  42. B(J+1)(J+2) J+1 BJ(J+1) J Absorption

  43. B(J+1)(J+2) J+1 BJ(J+1) J Emission

  44. General Relation for F(J) B(J+1)(J+2) J+1 F(J) BJ(J+1) J Harry Kroto 2004

  45. General Relation for F(J) B(J+1)(J+2) J+1 F(J) B(J+1) J BJ(J+1) J NB Common factor

  46. B(J+1)(J+2) J+1 F(J) B(J+1) J J F(J) = 2B(J+1)

  47. A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J =±1 E Transition Frequencies > F = 2B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc

  48. A Classical Description > E = T + V E = ½I2 V=0 B QM description > the Hamiltonian H J  = E J  H = J2/2I C Solve the Hamiltonian > Energy Levels F (J) = BJ(J+1) D Selection Rules > Allowed Transitions J =±1 E Transition Frequencies > F (J) = 2B(J+1) F Intensities > THE SPECTRUM J Analysis > Pattern recognition; assign J numbers H Experimental Details > microwave spectrometers I More Advanced Details: Centrifugal distortion, spin effect J Information obtainable: structures, dipole moments etc

  49. J 7 56B 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 0 0 Frequency

  50. J 7 56B 6 42B 5 30B 4 20B 3 12B 2 6B 1 2B 2B 0 0 2B Frequency

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