1. Lecture 6 Raman spectra of carbon nanotubes

2. Infrared (IR) spectroscopy IR 700 nm3500 nm400 nm Visible light IR IR spectra can be used to identify the types of molecules or functional groups

3. Energy Inter-nuclear distance (r) r eq Bond extensionBond compression Coulomb attraction Coulomb repulsion 2. Hooke’s law:  = -k(r – r eq )..(1)  : restoring force k: force constant E1E1 E2E2 1.E 1  E 2, vibration frequency remains the same but vibration amplitude increases (r  ). 3. Energy E = k(r – r eq ) 2 /2…(2) Diatomic vibration 4. The minimum energy (stable state) is when r = r eq, E = 0

4. A simple harmonic oscillator Vibration frequency  osc = (1/2  )  (k/  ) 1/2 Hz...(3)  : reduced mass of system  osc = (1/2  c)  (k/  ) 1/2 cm -1 ….(4) c: velocity of light This is Hz in unit, if one converts Hz into wavenumbers, Eq(3) can be expressed as Vibration energy is quantized E = ( + ½)ħ  osc (joules)…(5) = 0,1,2,3….(vibrational quantum number) Convert E into wave number unit, we have  = E /ħc = ( + ½)  osc (cm -1 )…(6) Eq(6) is energy allowed to a simple harmonic oscillator

5. Energy cm -l r eq Inter-nuclear distance (r) = 0  osc 1/2  osc 3/2  osc 5/2  osc 7/2  osc 9/2 = 1 = 2 = 3 = 4  = ( + ½)  osc Any molecules can never have zero vibration energy or atoms can never be completely at test relate to each other---(zero point energy E 0 = ħ  osc /2 (joules) or  o =  osc /2 (cm -1 ).

6. Vibrational changes due to interaction with radiation (electromagnetic wave) Vibrational changes can only be  =  1 (selection rules) + 1 - 1 + 2 - 2 wrong! Emission = ( + 1 + 1/2)  osc - ( + 1/2)  osc =  osc Adsorption =  osc Vibrational change always involves the same energy

7. 1.Electromagnetic wave only interacts (resonates) with molecules that can produce dipole moment. 2.Homogeneous molecules cannot produce dipole moment, so they do not have IR adsorption (e.g. N 2, H 2 ). Heterogeneous molecules can produce dipole moment so they have IR adsorption (e.g. CO 2, OH, cooH..) 3. For adsorption, the vibrating molecule only interacts (resonates) with electromagnetic wave at the same frequency.

8. How a molecule produces a dipole ? Example: water molecule (H 2 O) O HH - ++ dipole moment Net dipole = 0 O H H Linear form V-form Net dipole  0

9. Molecular dipole produced by vibrations A static linear H 2 O do not have a dipole But when molecule vibrates dipole may not be zero Symmetrical stretching Net dipole = 0 Asymmetrical stretching Net dipole  0

10. Time axis Net dipole IR radiation resonance

11. Types of vibration Symmetrical stretching Asymmetrical stretching

12. Scissoringtwisting wagging rocking Different vibrations give different frequencies (same molecules) C HH C H H

13. What if molecules have no intrinsic dipole, e.g. graphite No dipolar In this case, we induce dipole by laser beam (excitation)  Raman Raman is a name and he was a Indian

14. Laser beam

15. Raman spectrum of arbon nanotubes  20-30

16. High frequency region : D band: 1370 cm -1 (disorder structure) G band: 1580 cm -1 (graphitic structure) Low frequency region Radial breathing mode (RBM): < 300 cm -1

17. at higher frequency

18. A 1g mode at low frequency

19. g: Raman u: IR

22. I D /I G : determination of graphitization Smaller I D /I G  highly graphitization Larger I D /I G  less graphitization Why D-band means disorder structure and G-band for graphitized structure?

23. Different bond lengths

24. Bond length becomes roughly the same when approach center, but bond length remains great differences at periphery, so we can say That only one type of bond length at the center of the sheet and two types of bond lengths at periphery.

25. Such a C-C stretching motion mainly occurs at the central region of graphene sheets. E 2g mode is independent of sheet size and C-C vibration is locally. When size of graphene sheet increases the amount of vibration also increases, which leads to greater intensity of E 2g mode (G-band).

26. Such a C-C vibration is very sensitive to periphery regions, and is dependent of sheet size.

27. When sheet size increases, what happens? The central region increases, so amount of C-C stretching motion increases E 2g intensity increases. When sheet size decreases, what happens? The ratio between periphery and central regions increases. So amount of C-C stretching motion at central region decreases, and E 2g intensity decreases

28. When sheet size decreases, intensity of E 2g mode decreases and A 1g mode increases, why? and we said before that A 1g is sensitive to Size, why? Periphery region (two types of bond lengths) Central region (one type of bond length) because increases 1580 cm -1 1370 cm -1 1.42Å 1.36Å 1.46Å Small size of sheet Large size of sheet decrease

29. D-bandG-band I D /I G

33. 2 nm <

34. Various diameters of tubes 514 nm laser 613 nm shift

41. RBM also depends on temperature

42. Band shift to lower wave number is called softening, and shift to higher wave number is called hardening Why temperature increase causes softening to SWNTs

45. radial vibration C-C stretching C-C bending Thermal expansion of a tube C-C stretching undergoes the greatest influence by temperature variation

46. Why longer bond length gives lower vibration frequency?

47. 1. Inter-tube spring is a function of van der waals interaction 1 2 2. intra-tube spring is a function of C-C bond strength and tube diameter. C C R R 2 > 1