Achiral CNTs

All Aie achiral CNTs have a point of inversion symmetry located on the tube axis. For achiral CNTs with even n this can be easily demonstrated. In fact, for even n, Cc is a symmetry transformation, which together with cr acts as the inversion transformation I on CNT structure. 

Tire irreps are given as a function of the parameters k e [0, tt/ci] and m = -q, , /2]. The A = 0 condition means the suppression of all tire translations making the symmetry operations to collapse into a point group. This point group P is called isogonal and is given, for chiral and achiral CNT, by... 

General selection rules governing the process can be obtained by observing that the decomposition onto irreducible representations of the position operator both for chiral and achiral CNTs contains only contributions with k = 0 and m = 0,1. In fact, the matrix elements in Eq. (9) can be expressed in the Wigner-Eckart form as [5]... 

Let us recall, first, the full symmetry of CNTs, the identification of which has been completed recently by Damnjanovic et al. [23]. The non-symmorphic rod-group [23] describing achiral CNTs with index n can be decomposed in the following manner,... 

Fig. 2. 2D projection of various symmetries in achiral CNTs (armchair segment, top zigzag segment, bottom) is the primitive translation 82 is the screw axis with non-primitive translation and rotation, denoted by Tjl and C /2, respectively g is a glide plane and stand for the corresponding point group... 

The order N of the screw axis of an achiral CNT is equal to twice the order of its rotation axis n. Consequently, the SRs for the HHGS by achiral CNTs are... 

Figure 6.85. Illustration of an axial screw dislocation in a CNT. An achiral zigzag (n, 0) tube (a) can be viewed as a perfect crystal, and transformed into a chiral one by cutting, shifting by a Burgers vector (red arrows in b-d), and resealing a tube-cylinder (b). The free-energy profile shown in (e) illustrates the difference in growth of a chiral or achiral CNT with an increasing number of added carbon atoms. Reproduced with permission from Ding, F. Harutyunyan, A. R. Yakobson, B. I. PNAS 2009, 106, 2506.

It has been theoretically predicted that achiral SWNTs m = norm = 0) possess 8 RS-active and 3 IR-active modes, whereas chiral SWNTs 0 < m n) have 14 RS-active and 6 IR-active modes [10]. However, there are only few experimental reports on the IR-active phonon modes because of the difficulty in detection [40,41]. The difficulty is, in part, because SWNTs do not support a static dipole moment, and hence much weaker dynamic dipole moment has to be taken into account. Moreover, CNT samples usually contain carbon impurities that may induce similar IR absorption, making IR assigmnents much more difficult Under these circumstances, resonance HRS spectroscopy is expected to be useful for characterizing SWNTs. 

Figure 1. Symmetry of the. single wall CNT (8,6), (6,0) and (6.6). The horizontal rotational axes U and U are symmetrie.s of all the tubes, while the mirror planes (o , o),), the glide plane and the roto-reflectional plane are. symmetrie.s of the achiral tubes only (from [3])...

Within the formalism of the irreducible representation the position operator is decomposed in qFi + o o for chiral CNTs and E +o o for achiral topology. 

If the difference n-m is divisible by 3, then these CNTs have metallic properties. Semimetals are all achiral tubes such as chair . In other cases, the CNTs show semiconducting properties. Just t) e chair CNTs n = m) are strictly metal. 

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