Translation layer

Figure C2.2.4. Types of smectic phase. Here tire layer stacking (left) and in-plane ordering (right) are shown for each phase. Bond orientational order is indicated for tire hexB, SmI and SmF phases, i.e. long-range order of lattice vectors. However, tliere is no long-range translational order in tliese phases.

The Maier-Saupe tlieory was developed to account for ordering in tlie smectic A phase by McMillan [71]. He allowed for tlie coupling of orientational order to tlie translational order, by introducing a translational order parameter which depends on an ensemble average of tlie first haniionic of tlie density modulation noniial to tlie layers as well as / i. This model can account for botli first- and second-order nematic-smectic A phase transitions, as observed experimentally. 

Order and dense packing are relative in tire context of tliese systems and depend on tire point of view. Usually tire tenn order is used in connection witli translational symmetry in molecular stmctures, i.e. in a two-dimensional monolayer witli a crystal stmcture. Dense packing in organic layers is connected witli tire density of crystalline polyetliylene. 

It is not surprising that it is difficult to insert lithium between parallel layers which are randomly stacked. When lithium intercalates between AB stacked layers, a shift to AA stacking occurs [26]. It is likely that the turbostratically stacked layers are pinned by defects (which can only be removed near 2300°C ) preventing the rotation or translation to AA stacking. Thus, we can understand why varies as 372(1-P), the fraction of layers with AB registered stacking. More studies of the details of the voltage profiles in Fig. 7 can be found elsewhere [6,7,27]. 

Of particular importance to carbon nanotube physics are the many possible symmetries or geometries that can be realized on a cylindrical surface in carbon nanotubes without the introduction of strain. For ID systems on a cylindrical surface, translational symmetry with a screw axis could affect the electronic structure and related properties. The exotic electronic properties of ID carbon nanotubes are seen to arise predominately from intralayer interactions, rather than from interlayer interactions between multilayers within a single carbon nanotube or between two different nanotubes. Since the symmetry of a single nanotube is essential for understanding the basic physics of carbon nanotubes, most of this article focuses on the symmetry properties of single layer nanotubes, with a brief discussion also provided for two-layer nanotubes and an ordered array of similar nanotubes. 

Each of the membranes acts like a hard wall for dimer molecules. Consequently, in parts I and III we observe accumulation of dimer particles at the membrane. The presence of this layer can prohibit translation of particles through the membrane. Moreover, in parts II and IV of the box, at the membranes, we observe a depletion of the local density. This phenomenon can artificially enhance diffusion in the system. In order to avoid the problem, a double translation step has been applied. In one step the maximum displacement allows a particle to jump through the surface layer in the second step the maximum translation is small, to keep the total acceptance ratio as desired. 

Notwithstanding the large amount of work on pure iron and binary alloys, it remains difficult to translate the results to commercially useful steels. It is believed, on the one hand, that effusion of carbon monoxide can cause non-healing Assures in the scale , and on the other, that silicon creates self-healing layers at the metal interface . ... 

For a layer comprised of f grafted chains, the surface area, S, of the shell containing a given sublayer is given by S a fE,2. For a flat layer, all sublayers are of equal area and this translates to the Alexander result, t, (S/f)1/z d. However, for curved surfaces, S, and consequently depend on the distance, r, from the grafting site. Thus, a spherical layer is characterized by S r2, leading to E, a r/f1/2. For a cylindrical layer of length H, we have S rH and E, (rH/f)1/2. Once q(r)... 

To model the temporal behaviour of the turbulence induced aberrations we assume that a single layer of turbulence can be considered as frozen , but translated across the aperture by the wind. This is known as Taylor s frozen-flow hypothesis. The temporal behaviour can then be characterized by a time constant,... 

For smectic phases the defining characteristic is their layer structure with its one dimensional translational order parallel to the layer normal. At the single molecule level this order is completely defined by the singlet translational distribution function, p(z), which gives the probability of finding a molecule with its centre of mass at a distance, z, from the centre of one of the layers irrespective of its orientation [19]. Just as we have seen for the orientational order it is more convenient to characterise the translational order in terms of translational order parameters t which are the averages of the Chebychev polynomials, T (cos 2nzld)-, for example... 

---