Modern Polymer Spect..

3.5 Towards Lnrger Molecules: From Oligoiiiers to Polymers 101
fact that y is the phase shift of two equivalent oscillators belonging to two adjacent
chemical units within the crystallographic repeat unit (see above).
Following the same algebraic procedures as for the finite molecule the final
secular equation to be solved is [50]
where
GR(~) = (GR)’ + c [(GR)S eisq + (GR’)S ePisV‘] (3-44)
S
FR(~) = (FR)’ + c [(FR)~ eisv + (FR’)~ eCiSv]
S
(3-45)
Equation (3-43) is of 3pth degree in A. There are 3p characteristic roots (i = 1 . . .3p)
for each value of the phase y. The r = 3p functions i~,(v,) can be interpreted as
branches of a multiple valued function which is the dispersion relation for a onedimensional
polymer chain. The solution reached in Eq. (3-43) is analogous to that
previously attained by many authors in solid-state physics for simple tridimensional
lattices [51]. The important digerelice is that Eq. (3-43) treats the problem in internal
chemical coordinates, while, generally, simple lattices are treated in Cartesian
coordinates (Eq. (3-6)). The fhction is periodic with period ~ Tand C calculations are
limited to values of q~ within the first Brillouin zone [51, 521 (BZ) (-TC I y i TC). In
most cases, the results for only half of the BZ are reported, the second half being
symmetrical through y z 0.
In the spectroscopy of polymers as one-dimensional ID lattices it is generally
easier to deal with the phase shift y at odds with solid-state physics, dealing with
three-dimensional 3D crystal, which treats the whole dynamics in terms of the
vector k. For ID lattices we can describe the wave-motion (phonon) propagating
along the 1D chain either by the phase shift v, or by the vector Ik/ = y/d where k
has only one component along the chain axis k,. In contrast, in the spectroscopy
and dynamics of 3D lattices (where intermolecular forces are active in all directions)
phonons are labeled with k vectors with three components (kk, k,, k,) [51]. When k
is used in polymer dynamics Eq. (3-43) can be easily rewritten as
It should be remembered that in a single and isolated polymer, chain atoms
are allowed to move in the tridimensional space even if phonons are considered
to propagate in one direction along the chain axis. This means that no neighboring
intermolecular interactions are taken into account, i.e., the dynamics is that of a
chain ‘in uacuo’.