Modern Polymer Spect..

3.6 From Dynamics to Vibrational Spectra of One-Diniensiorzal Lattices 109
translational (or rototranslational) symmetry element is properly taken into account
(in combination with point-group symmetry operations) a factor group isomorphous
to a symmetry point group can be defined and k = 0 phonons can be separated
into symmetry species, just as in the case of finite molecules. Traditional
infrared and/or Raman activities can be predicted as well as the optical transitions
associated to specific dipole allowed transitions with their directional properties for
oriented samples.
The physical meaning of the selection rule which states that only k = 0 phonons
can be observed spectroscopically is that the atomic displacements in translationally
equivalent crystallographic units must be all in-phase. When the geometry of the
polymer chain (i.e. $(8, 1) and t = no. of turns of the helix within the rototranslational
unit cell) is taken into account, phonons are infrared active for y = 0 and
y z 0 (doubly degenerate); Raman activity is achieved for phonons with y = 0
(nondegenerate), 0 and 20 (both doubly degenerate) [15, 61, 621. Degeneracy is
clearly understood when folding of the dispersion curves is carried out at the zone
boundary in going from y to k space. If p is the number of atoms in the chemical
repeat unit for a helical polymer, one expects in the infrared 3p - 2 totally symmetric
11 modes with q = 0 and 3p - 1 doubly degenerated I modes with y = 0.
Coincidences with the infrared spectrum are expected in the Raman with the modes
at y = 0 and y = 8, while extra modes are expected at y = 28 (practically never
observed because their Rainan intensity is very weak).
Let us further consider the direction of the transition dipole moments in infrared
for a stretch-oriented polymer sample in which it is assumed that polymer chains
are aligned along the stretching direction. Phonons with 9 = 0 have their transition
dipole moment parallel to the chain axis which coincides with the direction of
stretching (11 modes); modes with y = 8 have transitions moments perpendicular to
the chain axis (1 modes). They can easily be identified in an infrared experiment on
stretch-oriented samples carried out in polarized light. Analogously, the elements of
the polarizability tensor can be selectively excited by Ranian scattering experiments
on oriented polymer samples examined in polarized light in diflerent scattering
geometries. For a study of the Raman spectrum of a stretch-oriented polyethylene
rod in various geometries see [28] and Figure 3-7 [63].
A few comments of general relevance need to be made.
The shape of the dispersion curves and the corresponding eigenvectors which
describe the vibrational displacements (phonons) depend necessarily on the vibrational
force field adopted in the calculations. The fitting of the experiniental frequencies
with the calculated ones, although being a good indication that the force
field is reasonably acceptable, can by no means be taken as proof of its general
reliability throughout the whole BZ. Indeed, fitting has been obtained only at two
values of the phase coupling y. The intramolecular interactions may be different at
y # 0 and y # 8 or 20, thus in principle changing the shape of the theoretically
predicted dispersion curves (see. for example, [64]). Other independent experiments
are needed before claiming the success and generality of the intramolecular force
field.
So far, we have discussed k = 0 phonons of the one-dimensional lattice and have