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My ABC System of Spectroscopy

Explore the fundamentals of rotational spectroscopy, including the Hamiltonian, energy levels, selection rules, allowed transitions, and spectrum analysis for linear molecules. Discover advanced details like centrifugal distortion and spin effects.

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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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