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Extra info for Carbon Nanotubes: Basic Concepts and Physical Properties
The Raman active modes in carbon nanotubes are then + Raman active: 2 4 I, 3 El, 2A1, + 3E2, + 2E1, +4E2, 3A1+ 5E1+ 6Ez or 20~: + 30E; + 3 0 ~ : or 2oAt + 2oE; or 3(& + 50EI + 60Ez +4"~: (zig-zag) (armchair) (chiral tubcs). 38) The two AI, phonons arc the radial breathing mode, where all carbon atoms move in-phase in the radial direction, and lhe high-cncrgy longitudinal (zig-zag) and transverse (armchair) modes. In chirid tubes, both the longitudinal and the transverse phonon are fully symmetric (All In Sect.
Thc situation is different in zigzag tubes, where axial modes can have At, symmetry. 2 26 Structure and Symmetry e . g , the Csand I C : ~pair. An axial phonon eigenvector is, therefore, never of A,, symmetry in armchair carbon nanotubes. Indeed, the axial high-energy mode is of At, symmetry and hence not Raman active. As a lasl example we project a circumferential displacement with the help of the P I(E2,) (,) operator. We use the E , C;,and ox symmetry operations to demonstrate projection operators onto degenerate representations.
As another example we also know the rn quantum number of the confined electronic states in a 7one-folding calculation. In Fig. 7 the bands with m = f9 are indicated. Optical transitions for light polarized along the nanotube axis arc only allowed for Am = 0, see Chap. 4. 5eV from Fig. 7. Therc is one concept that beats the zone-folding approximation in spced and physical insight. It is based on the line group symmetry of carbon nanotubes. Since this approach is conceptually quitc involved, we will not go into dctails in this book.