Modern Polymer Spect..

120 3 Vibratiorial Spectra us a Probe of Structural Order
The following observations should be kept in mind.
1. Because of the shape of the calculated v(q), which may show many maxima and
minima (due to the vibrational coupling intrinsic to the system and to the force
field used), g(v) will also show corresponding strong or weak singularities. Some
of these singularities necessarily coincide with k = 0 spectroscopically active
phonons, while other will not be seen in the optical spectrum of a perfect system,
but will (in principle) be observed in the neutron-scattering spectra and possibly
in the optical spectra of a partially disordered polymer. In a disordered material,
no symmetry selection rules are active and all modes may gain some activity.
The vibrational spectra of a disordered material can thus be considered the
mapping of g(v) dipole (or polarizability) weighted. The g(v) calculated with
NET or any other numerical or analytical method is, instead, dipole (polarizability)
unweighted. Any comparison with the experimental g(v) must be made
with great caution.
2. As discussed in Section 3.5 for tridimensional lattices, v i 0 for k 40 for the
three acoustical branches with a positive slope and with a discontinuity at k = 0
(r). The shape of g(v) i 0 has been matter of strong interest in physics for the
calculation of specific heat and related thermodynamic quantities. As already
mentioned for one-dimensional lattices in LIUCUO, some of the acoustical branches
tend asymptotically to zero. The derived calculated g(v) shows a very strong
singularity at v = 0. Such a singularity is meaningless and only due to the limitation
of the molecular modes adopted in the calculations (1D lattice) which is
unable to account for intermolecular forces. Also, for real polymer samples the
experimental g(v) i 0 for k i 0 as expected for classical crystals since they do
interact with the neighboring chains with very weak intermolecular forces.
3. Additional singularities both in calculated and experimental g(v) are expected at
very low energies (~0-30 cm-’) originating from very low-frequency 3D-lattice
phonons. These singularities may coincide with the optical phonons in infrared
and/or Raman spectra at very low frequencies.
As already anticipated, a complementary experimental technique for deriving
information on the dynamics (frequencies and vibrational amplitudes) of polymers
or of materials in general is the use of inelastic neutron-scattering techniques (INS).
After a long development time, during which experiments were difficult and provided
limited information, the instruments in a few specialized centers recently
began to provide detailed data covering the whole spectrum. Thus, we predict a
‘renaissance’ of INS techniques for the studies of molecular and lattice dynamics.
For a thorough discussion of INS spectroscopy we refer the reader to specialized
publications [8 13 (see also other references quoted below); here, we restrict ourselves
to the basic principles of the technique connected especially with the dynamic
quantities which are of interest in optical spectroscopy.
Let a beam of cold neutrons made mono-energetic with wave-vector kll be shone
onto a sample, and let the beam be scattered inelastically and incoherently. The
outgoing neutrons have wave-vector kf and are collected and detected by suitable
devices. Since the energy of the thermal neutrons is of the same order as that of