Modern Polymer Spect..

3.8 Driisity of Vibratioiid StLitrs arid Neutron Scuttering 121
lattice and molecular vibrations, neutrons impinging on the sample may gain or lose
energy in a way proportional to g(v).
It has been shown [S2] that the scattering cross-section of a polycrystalline material
is approximately the same as that of cubic crystals. We are, at present, concerned
with one-phonon processes and neglect the possibility that multiphonon
processes may occur (also in vibrational spectra we have neglected anharmonic
effect with the consequent appearance of overtones, combinations, and hot bands).
The one-phonon scattering cross-section for a cubic lattice can be written as [83,
S4]:
d2a,,,,h/d0 do = A1 kf 1 /I ko lep2w (hK2/2M)[g(co)/oi]
x [h'(of - wo + o)/(eP" - I)] (3-51)
In Eq. (3-51), and in a few equations which will follow the vibrational frequencies
are not expressed as v in wavenumbers, but as w in Hz. In Eq. (3-51), A is the
scattering cross-section of the scattering nucleus (since the H atom has a very large
cross section, namely AH = 82.5 barn, Ac = 5.5 barn, molecules containing hydrogen
are very suitable for neutron experiments), lkfl and lkol are the moduli of the
wave-vectors of the scattered and incident neutron respectively, e-2w the Debye-
Waller factor or temperature factor, K = kf - ko, M the mass of the unit cell, g(w)
the desired density of vibrational states, and p = h/?kBT (where kB and T are the
Boltzman factor and the temperature respectively).
It is clear that thermal population is strongly contributing to the scattering crosssection
with the Debye-Waller factor and the term p. Measurements must then be
carried out at very low temperature.
The experimental g(co) (or g(v)), while showing qualitatively all the features of
the calculated g(w), cannot yet be quantitatively compared because the vibrational
amplitudes (generally called polarization vectors in neutron-scattering circles) of the
various normal modes (phonons) are not yet taken into account. The situation is
similar to the features of the vibrational spectra of an isotropic sample compared
with the features of the spectra in polarized light of an oriented anisotropic sample.
Let us take a stretch-oriented sample of a polymer subject to a neutron beam with
incident wave-vector ko. A more complete expression of the scattering cross-section
is the following [S2, 841:
Some of the terms appearing in Eq. (3-52) are already defined for Eq. (3-51). The
new terms are the following: A, is the scattering cross-section of the n-th nucleus of
the unit cell, e-2W11 the Debye-Waller factor for the n-th nucleus of mass MI,, coi the
frequency of the j-tli normal mode of the r-th branch in the dispersion relation. The
most important quantity is ek. C:' which gives the projection along ko of the atomic