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The complex number e is elmil/N, so that e' = exp(-2mil/N).

The complex number e is elmil/N, so that e' = exp(-2mil/N).

tations are related to the Nth roots of unity. The number of classes and irreducible representations of group CN/Sl is thus equal to N, counting each of the partners of the Ei representations as distinct. Various basis functions for group CN/n that are useful for determining the infrared and Raman activity of the chiral tubules are listed in Table 19.5. We note that the irreducible representations A and Ex are infrared active and A, Ex, and E2 are Raman active.

For tubules where (n, m) have a common divisor d, then after N translations t, a length Td is produced, and il unit cells of area (ChT) are generated. For d ^ 1 the character table for the Abelian group in Table 19.6 contains 1 /d as many elements, so that the order of the Abelian group becomes N/d. This case is discussed further below.

The space group for a chiral nanotube specified by (n, m) is given by the direct product of two Abelian groups [19.82,87]

where d' is the highest common divisor of il/d and d, while d is the highest common divisor of (n, m). The symmetry elements in group Cd, which is of order d' include

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