Carbon Material with a Highly Ordered Linear-Chain Structure - Polyynes: Synthesis, Properties, and Applications - page 226

the IR and Raman examination for the correct assignment of the calculated
vibration modes should also be taken into account.
The high-frequency band in the region of 2000 cm 1 is commonly attrib-
uted to stretching vibrations frequency, of long sp 1 -hybridized carbon

chains, which can be approximately calculated from the following equation:

!

1 = 2

! ¼ð 4k = m Þ

where k is an elastic constant, m is mass of the carbon atom. The other
peak at 1550-1580 cm 1 corresponds to the position of the sp 2 -carbon
bond vibration frequency and, therefore, it is conventionally considered to
be a feature of sp 2 -carbon phase. However, this peak can alternatively be
explained by the presence of regular kinks in the polycumulene carbon

chains [2]. Figure 11.12 shows a calculated density of phonon states in the

polycumulene carbon chain where elastic constants were chosen to be equal
to k 1 in linear fragments of the chain and to k 2 in the region of the kink; the
k 2 /k 1 value is varied from 1 (Figure 11.12(a)) to zero (Figure 11.12(e)). In
these calculations only one component of the atomic coordinates was taken

into account (i.e. along the chain axis). As can be seen from Figure 11.12 ,

the phonon band of cumulene (a) transforms to discrete frequencies of the
butatriene molecule when k 2 ¼ 0 (e) due to the alternation of the k constant.

Besides that, every sub-band has two maximums at the threshold values,
which are a specific feature of the one-dimensional system. This may be

FIGURE 11.12 Calculated carbon chain phonon density of states for k 2 /k 1 ¼ 1.0 (a),
0.9 (b), 0.6 (c), 0.4 (d), 0 (e).