Brillouin zone polyethylene

In principle, crystalline polymers will yield to the same types of analyses. In practice, significant dispersion, even in the internal modes, is often present and calculations across the complete Brillouin zone are needed. These are still rare even for the simplest systems such as polyethylene, but are an obvious next step. 

The internal modes of typical molecular crystals show little dispersion and a calculation of the frequencies at the F point in the Brillouin zone is usually sufficient for good agreement between observed and calculated spectra. However, there are many examples where significant dispersion is present the alkali metal hydrides ( 6.7.1), graphite ( 11.2.2) and polyethylene ( 10.1.1.1) being notable cases. In these instances, a calculation of the full dispersion curves are needed. In... 

However, it is important to note that polymer quantum chemistry is not a ID, solid-state physical science. In strictly ID physics, the systems are periodic in ID and have ID wave functions. In polymer quantum chemistry, the systems and their wave functions are 3D but periodic only in ID. Usual theorems of ID physics are consequently no longer valid [28]. A typical example is that of extrema of the energy bands, which should only occur at the center and the edges of the Brillouin zone in a strictly ID system. For polymers, even in simple cases like the linear zigzag polyethylene chain, some extrema of the energy bands are encountered at arbitrary positions in the first Brillouin zone that are not points of high symmetry (Fig. 36.3). 

The absorption spectrum of polyethylene in the far infrared is very simple a broad background increasing to higher frequencies and only one sharp band at 73 cm. Figure 15 shows an example for a high density polyethylene (HDPE). The background comes from the amorphous part of the material, which can be described in terms of the variable Brillouin zone. The absorption band itself, called the "73 cm band," is generated by a lattice vibration of the orthorhombic lattice, which is shown in Fig. 16. 

FIG. 4 Orthorhombic unit cell of polyethylene, with 50% probability thermal vibration ellipsoids for each atom, computed using the force field of Ref 55 with 8000 normal mode vibrations sampled in the Brillouin Zone. Thermal vibration ellipsoids were plotted using Ortep-III [48]. 

The Brillouin zone for polyethylene is shown in Figure 4. The FUC of polyethylene consists of the unit CaH4, Each carbon atom donates four valence orbitals and four electrons, and each hydrogen donates one orbital and one electron, so there are ten orbitals and ten electrons. These form ten energy bands, five of which are filled. 

Recently, Brillouin scattering has proved useful in this area for studying the frequency dependence of hypersonic (GHz zone) absorption and dispersion velocity in liquid sulphur dioxide [91] the effect of isotopes on hydrodynamic fluctuations in self-associated fluids [92] and the elastic properties of polyethylene glycol solutions in water, benzene and toluene [93]. 

---