Rectangular lattice

S. Chains in the S phase are also oriented normal to the surface, yet the unit cell is rectangular possibly because of restricted rotation. This structure is characterized as the smectic E or herringbone phase. Schofield and Rice [204] applied a lattice density functional theory to describe the second-order rotator (LS)-heiTingbone (S) phase transition. 

Figure BT2T4 illustrates the direct-space and reciprocal-space lattices for the five two-dimensional Bravais lattices allowed at surfaces. It is usefiil to realize that the vector a is always perpendicular to the vector b and that is always perpendicular to a. It is also usefiil to notice that the length of a is inversely proportional to the length of a, and likewise for b and b. Thus, a large unit cell in direct space gives a small unit cell in reciprocal space, and a wide rectangular unit cell in direct space produces a tall rectangular unit cell in reciprocal space. Also, the hexagonal direct-space lattice gives rise to another hexagonal lattice in reciprocal space, but rotated by 90° with respect to the direct-space lattice.

Figure C2.2.7. Schematic illustrating tire classification and nomenclature of discotic liquid crystal phases. For tire columnar phases, tire subscripts are usually used in combination witli each otlier. For example, denotes a rectangular lattice of columns in which tire molecules are stacked in a disordered manner (after [33])...

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

X-ray diffraction data relating to the ditholium salt shown in (b) for M = 12. A and B correspond to the rectangular lattice vectors shown in D and a and b correspond to the N,-q to Dh and to D, phase transitions. Reproduced from reference 30 with permission. 

Derrick A semipermanent structure of square or rectangular cross-section having members that are latticed or trussed on all four sides. This unit must be assembled in the vertical or operation position, as it includes no erection mechanism. It may or may not be guyed. 

Fig. 7. Two-dimensional packing observed for the main-chain polymer with two odd numbered spacers of different length. Solid lines and arrows indicate the two-dimensional rectangular lattice and macroscopic polarization, respectively (Watanabe et al. [68])...

Fig. 11a,b. Examples of one and two-dimensional structures for terminally fluorinated mesogens a antiparallel dimeric layering for the compounds with one terminal fluorinated tail (Diele et al. [38]) b two-dimensional (columnar) packing observed for the swallow-tailed fluorinated compounds solid lines indicate two-dimesional rectangular lattice (Lose et al. [41])... 

FIG. 7 Uniform-pore models (figure based on that of Ref. 276). (a) Cubic lattice. (From Ref. 276.) (b) Conical lattice. (From Refs. 300 and 301.) (c) Spherical lattice. (From Ref. 297.) (d) Cylindrical lattice. (From Ref. 3.) (e) Circular pores in rectangular sheets. (From Ref. 3.) (f) Rectangular pores. (From Ref. 364.)... 

In two dimensions, five different lattices exist, see Fig. 5.6. One recognizes the hexagonal Bravais lattice as the unit cell of the cubic (111) and hep (001) surfaces, the centered rectangular cell as the unit cell of the bcc and fee (110) surfaces, and... 

The discotic mesophases are classified in two types columnar, and nematic discotic. The structure of the nematic discotic mesophase (Np, Figure 8.3, left) is similar to that of rod-like molecules, but constituted by disk-like units. In columnar mesophases, the molecules are stacked in a columnar disposition and, depending on the type of columnar arrangement, several columnar mesophases are known. The most common lattices of the columnar phases are nematic discotic (No), columnar nematic (Ncoi), columnar hexagonal (Coin), and columnar rectangular (Col,) mesophases. 

In two dimensions, the nodes occupy the vertices of a regular lattice, which is usually rectangular (Figure 3.5). This layer of nodes is sometimes known as a Kohonen layer in recognition of Teuvo Kohonen s (a Finnish academician and researcher) work in developing the SOM. 

If the input data are not spread evenly across the x/y plane, but are concentrated in particular regions, the SOM will try to reproduce the shape that is mapped by the input data (Figure 3.23), though the requirement that a rectangular lattice of nodes be used to mimic a possibly nonrectangular shape may leave some nodes stranded in the "interior" of the object. 

The behavior of CA is linked to the geometry of the lattice, though the difference between running a simulation on a lattice of one geometry and a different geometry may be computational speed, rather than effectiveness. There has been some work on CA of dimensionality greater than two, but the behavior of three-dimensional CA is difficult to visualize because of the need for semitransparency in the display of the cells. The problem is, understandably, even more severe in four dimensions. If we concentrate on rectangular lattices, the factors that determine the way that the system evolves are the permissible states for the cells and the transition rules between those states. 

Suppose that a single cell in a rectangular lattice has the state ON in an environment in which all other cells are OFF. The transition rule is... 

A. Patrykiejew, S. Sokolowski, K. Binder, Phase transitions in adsorbed layers formed on crystals of square and rectangular surface lattice, Surf. Sci. Reports 37, 207 (2000). 

---