Single crystals unit cell

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

The rather large estimated standard deviations of the cell constants reflect changes during data collection, presumably as a consequence of some decomposition. A powder photograph of the bulk material was indexed using the single-crystal unit cell dimensions. 

Figure 7. Structure of the 6T single crystal unit cell [61]. The a, b and c-axes are indicated as well as the molecular geometry long in-plane axis (L), short in-plane axis (M) and (N) perpendicular to L and M.

It is important to note that no motion having a period in excess of L/v can be reproduced in the simulations, where L is the length of the simulation box and is a velocity of sound in the medium.In addition, use of periodic boundary conditions together with a single structural unit cell as the simulation box restricts the calculation of spectral quantities to those at the center of the Brillouin zone the periodic boundary conditions force atoms in all images of the simulation box to vibrate in-phase, that is, the definition of a motion at the center of Brillouin zone. When comparing results of the calculations with the experimental spectra, one must also bear in mind that the model used in the calculations implies a perfect crystal structure, whereas experiments are usually done with microcrystals having defects. 

Each of these terms is identical to the structure factor as described in Equation (14), except that the summation is over a subset of atoms. Fuc is summed only over atoms within a single bulk unit cell, Fsurf is summed over all near-surface atoms that might be displaced from their ideal bulk lattice positions (typically 2-3 layers deep into the crystal) plus any adsorbed layers attached to the surface, and Fwater describes the fluid structure above the interface, including any structuring of the fluid near the mineral surface. Equation (16a) can be rewritten to express the scattering intensity,... 

If the structure factor for an ideally smooth surface, Fldeai, is written by summing laterally over a single surface unit cell and vertically over all z (including the semiinfinite sum over the bulk crystal, the sum over the surface layer and any fluid layer above the surface) as written in Equation (16), the structure factor for a rough surface can be written ... 

For preciseness, let us begin by considering an aggregate of crystallites, i.e. an aggregate of small regions each defined by a crystal unit cell. We then ask how we would define the orientation of a single imit cell of lowest symmetry, i.e. a triclinic unit cell. 

Another unique characteristic of the rigid-rod fibers is that they are single-phase paracrystalline fibers, which is significantly different from many polymer fibers with a two-phase semicrystalline structure. Table 5 lists refined PBZT, PBZO, and PIPD crystal parameters. All of them have monoclinic crystal unit cells. For PBZT, two molecular chains pass through an oblique unit cell. Figure 15 shows the crystal structural models of PBZT suggested by Takahashi (218) and Fratini (217). In the Takahashi crystal model of PBZT, there is disorder... 

Disregarding any detailed morphological models proposed for the lamellar crystals in semicrystalline polymers in the past half centiny, crystal imit cells in semicrys-talUne poisoners exactly obey the rules of classical crystallography, which consist of 7 crystal systems and 14 Bravais lattices (more precisely, there are only 6 crystal systems in polymer crystals, and the cubic system does not exist). Most of the unit cell determinations of the semicrystalline polymers rely on the wide-angle x-ray diffraction (waxd) experiments on oriented poisoner fibers and films. This is be-canse of small crystal sizes in polymers, which lead to a difficult experimental task to obtain single-crystal waxd results on semicrystalfine polymers. It is also possible to nse electron diffraction (ed) method in transmission electron microscopy (tern) to determine polymer lamellar crystal unit cell structures, dimensions, and symmetries. 

The best method for preparation of single crystals of polymers is to grow them from dilute solution. Usually such crystals are small, typically a few micrometres across, and are plate-like with a regular shape that reflects that of the crystal unit cell. The thickness is typically 10 nm. By analysis of electron diffraction patterns from single crystals of polyethylene (the first polymer to be crystallized in this way), it was shown that the polymer... 

Note that the acoustic branch corresponds to the oscillation of the crystal unit cells relative to each other, whereas the optical branches describe the oscillation of atoms relative to each other within the volume of a single unit cell. 

When atoms, molecules, or molecular fragments adsorb onto a single-crystal surface, they often arrange themselves into an ordered pattern. Generally, the size of the adsorbate-induced two-dimensional surface unit cell is larger than that of the clean surface. The same nomenclature is used to describe the surface unit cell of an adsorbate system as is used to describe a reconstructed surface, i.e. the synmietry is given with respect to the bulk tenninated (unreconstructed) two-dimensional surface unit cell. 

---