Modern Polymer Spect..

lowing. The CHl-bending mode 6 [certainly of A1 symmetry ( 1 1 ) )
3.17 Fermi Resonaiices 167
(Figure 3-37b)
observed at 1460 cm-~' should have its first overtone 126 : 2 x 1460 = 2920 cni-',
A1 x A1 = Al) which can enter Fermi resonance with the d+ mode of A1 symmetry.
Eigenstates mix, frequencies are split apart with intensity borrowing froin the
fundamental to the overtone producing two strong lines at 2876 and 2942 cni- '
(Figure 3-38a-c'i.
It should be remembered that the first overtone 26 should generally have vanishing
intensity, but because of Fernii resonance its frequency is pushed upward and
it borrows a lot of intensity from the fundamental d+ observed at 2876 cm-' .
A similar, but seemingly more complex case, occurs for polymethylene chains.
Let us consider the dispersion curve of polyethylene or the corresponding vibrations
of finite polymethylene chains (Figure 3-39 j whose frequencies lie on these dispersion
curves at finite values of the phase coupling (Section 3.5). The vibrations of the
lower portion of the dispersion branch of CH2 rocking have their first overtone
levels ( - 720 x 2 zz 1440 cnrl) close to the fundamental levels of the CHZ-bending
motions. On their turn, the first overtones of the levels along the dispersion curve of
the CH2-bendings occur just in the frequency region where the fhdaniental CH
stretching modes occur (2 x -1440 z 2880 cnirl). When symmetry allows, Fernii
resonances occur and levels repel each other with drastic intensity changes. The
issue becomes formally more complex since (Section 3.6) when molecules crystallize
in an orthorhonibic lattice (with two molecule per unit cell) the number of frequency
branches doubles, thus doubling the number of levels which can generate
both overtones and combinations which may produce Fermi resonances in the
proper frequency ranges, as just indicated.
These Fernii resonance phenomena have found detailed mathematical treatment
first tackled by Snyder et al. [138, 1391 for single all-tram chains and later reformulated
by Abbate et al. for ii) single chains, (ii) an orthorhombic lattice [140], (iii)
all-trans, and (iv) conformationally distorted chains [ 141, 1421. The most interesting
features to be accounted for are the Raman scattering of polymethylene chains in
the frequency regions centered near 1450 and 2900 cnirl.
For the sake of simplicity in the discussion which follows we label with (+) and
(-) the symmetric and antisymnietric combinations respectively either of internal
coordinates (e.g., C-H stretches within a CH? group) or of 'group coordinates' between
two CH2 units within the 'chemical repeat unit' made up by two CH? groups.
First, we discuss the case of true k = 0 modes for an infinite chain (or the case of
k 4 0 for finite long chains). Let us, for the sake of simplicity, consider the spectrum
of traiis polyethylene both as a single chain and crystallized in an orthorhombic
lattice with two molecules per unit cell. Since the spectrum of polyethylene
as single chain is not available for this discussion we necessarily use the calculated
spectra as simulation of reality [140]. Let us first compare the general features of the
calculated Raman spectra in the CHz-bending centered near 1435 cm-' for a single
chain and for crystalline polyethylene (Figures 3-40a, b) (point group D?!,). The
single very strong line near 1435 cm-' (Figure 3-39) is to be assigned to the limiting
k = 0 totally symmetric in-phase CH?S+ mode of A, species. For the crystalline
material, such a mode splits into two levels near 1420 (Ag) and 1435 an-' (BI?); an
additional broad scattering (with a wing towards higher frequencies) is calculated