Modern Polymer Spect..

3.5 TOIIYW~ Laiyer Molertiles: Fvoiii Oligomevs to Polimers 103
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Figure 3-2. Dispersion curves \./a, for single chain helical
(z = IS, t = 7) polytetrafluoroethylene.
0' 60' 120" lBi
plot of v(q) becomes usable while v(k) becomes unreadable (117 frequency phonon
branches in k space) (Figure 3-2) [54].
Dealing with 1D chains iii vuczio carries an important consequence on the shape
of the dispersion curves if compared with the same systems considered as 3D
lattices. From lattice dynamics in 3D it is known that if q is the number of atoms
in the crystallographic repeat unit the dispersion relation provides 3q phonon
branches. three of which are labeled as 'acoustical' and 3q - 3 as 'optical' branches.
The former three have zero frequencies at the so-called r symmetry point in k space
(i.e., with k, = k, = k, = 0; see Figure 3-3) and correspond to the three nongenuine
vibrations with zero frequencies which describe the three rigid translations of the
whole lattice along the three Cartesian coordinates [49. 551. The 3q - 3 optical
branches may interact with the electromagnetic waves according to specific selection
rules and provide the system with particular optical properties (absorption and/
or Raman scattering), as discussed below.
In ID systems iii u~~cuo the rigid rotation of the whole chain about its axis is not
hindered. It follows that an additional zero eigenvalue is expected from Eq. (3.43)
when applied to 1D systems at k = 0. The lack of intermolecular forces for the
polymer chain in uact[o carries an important consequence on the shape of the two
acoustic branches near k = 0. While the three acoustic branches for 3D crystals
have a discontinuity at k = 0, for ID crystals two of the acoustical branches may
approach ~(0) almost asymptotically [56]. It follows that the dynamics of 1D systems
based on the model of an isolated chain at small k values and at very low
frequencies is totally wrong and cannot be used for the interpretation of related
physical properties of real systems (neutron-scattering experiments, thermodynamic
properties, Debye Waller factors, mean square amplitudes, and other physical
properties where thermal population is involved which is strongly affected by lowenergy
vibrations).
For the benefit of the discussion which follows we give here a calculated schematic
example of the procedures followed in setting up the matrices F~(rp) and