Two-dimensional lattices

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... 

The behavior of an adsorbate on a single patch of size L has been represented by the familiar two-dimensional lattice gas model Hamiltonian with the added term resulting from the presence of a boundary field ... 

Just as was the case for one-diinensional majority rules considered in the previous section, we again recognize that the two-dimensional majority rule is but a special case of the generalized threshold rule defined in equation 5.121. Intuitively, the idea is simply to let aij represent a two-dimensional lattice that is built out of our a-priori structureless set of sites, i = 1, 2,..., A. Suppose we arrange these N sites into n rows with rn sites per row, so that N = n x rn. Then the site positioned on the row and column, can be identified with the original site... 

Now consider an entire temporal history of this PCA. That is, consider the effective two-dimensional lattice that is formed by stacking successive one-dimensional layers on top of one another (see figure 7.4). Because of the Markovian nature of the evolution, the probability of this temporal history is given simply by... 

Figure 12.10 shows a small section of a two-dimensional lattice and its dual lij is the length of the link between sites i and j, while lij is the border length of their dual cells. More generally, in n-dimensions, bj is the volume of the (n - 1)-dimensional surface in the dual lattice. In terms of these variables, Lee defines the weights Xij to be equal to... 

For each of the two-dimensional arrays shown here, draw a unit cell that, when repeated, generates the entire two-dimensional lattice. 

These structures were firstly observed for terminally polar mesogens [11, 12]. However, recent experiments give clear evidence of the presence of smectic A layering [37, 38], re-entrant nematic behaviour [39], two-dimensional lattices [40, 41] and smectic layering with incommensurate periodicities [42] for non-polar sterically asymmetric LCs. 

The steric frustrations have also been detected in LC polymers [66-68]. For example, the smectic A phase with a local two-dimensional lattice was found by Endres et al. [67] for combined main chain/side chain polymers containing no terminal dipoles, but with repeating units of laterally branched mesogens. A frustrated bilayer smectic phase was observed by Watanabe et al. [68] in main-chain polymers with two odd numbered spacers sufficiently differing in their length (Fig. 7). 

The common periodic structures displayed by surfaces are described by a two-dimensional lattice. Any point in this lattice is reached by a suitable combination of two basis vectors. Two unit vectors describe the smallest cell in which an identical arrangement of the atoms is found. The lattice is then constructed by moving this unit cell over any linear combination of the unit vectors. These vectors form the Bravais lattices, which is the set of vectors by which all points in the lattice can be reached. 

FIG. 6 The three types of two-dimensional lattices of hydrocarbon chains. 

A major limitation of diffraction techniques has been the need to obtain crystalline samples. If scientists could learn how to crystallize large molecules in a routine manner, a breakthrough would result. In the biological area, this limitation is keenly experienced for membrane-bound proteins, which are important in many biological functions. Scientists are now devising techniques and strategies to crystallize these proteins—if not in three-dimensional, then in two-dimensional lattices. 

Miyazaki, H. T. Miyazaki, H. Ohtaka, K. Sato, T. 2000. Photonic band in two-dimensional lattices of micrometer-sized spheres mechanically arranged under a scanning electron microscope. /. Appl. Phys. 87 7152-7158. 

Analyzing orientational structures of adsorbates, assume that the molecular centers of mass are rigidly fixed by an adsorption potential to form a two-dimensional lattice, molecular orientations being either unrestricted (in the limit of a weak angular dependence of the adsorption potential) or reduced to several symmetric (equivalent) directions in the absence of lateral interactions. In turn, lateral interactions should be substantially anisotropic. 

Chapter 3 is devoted to dipole dispersion laws for collective excitations on various planar lattices. For several orientationally inequivalent molecules in the unit cell of a two-dimensional lattice, a corresponding number of colective excitation bands arise and hence Davydov-split spectral lines are observed. Constructing the theory for these phenomena, we exemplify it by simple chain-like orientational structures on planar lattices and by the system CO2/NaCl(100). The latter is characterized by Davydov-split asymmetric stretching vibrations and two bending modes. An analytical theoretical analysis of vibrational frequencies and integrated absorptions for six spectral lines observed in the spectrum of this system provides an excellent agreement between calculated and measured data. 

Ferroelectric ordering in certain infinite two-dimensional lattices is due to the long-range contribution of dipole forces. Thus, it is not surprising that in limited two-dimensional lattices numerical calculations of dipole interactions lead to the replacement of ferroelectric states with macrovortex states64 which approximate to ferroelectric states far from the center of the limited lattice (coinciding with the center of the macrovortex). 

Anisotropic dipole interaction in two-dimensional lattices with a no higher than the second-order symmetry axis leads to the fact that the function J(k) does not go to zero at any k. Actually, in this case / = 1, P h, 12) > Pi(h, 12) in Eq. (2.2.14) and... 

If a ferroelectric ground state is realized on a two-dimensional lattice with a symmetry axis of order higher than two (as with a triangular dipole lattice), the... 

There is another case when the orientational Hamiltonian for nonpolar molecules (2.3.1) on a symmetric two-dimensional lattice is reducible to a quasidipole form, viz. the case of planar orientations of long molecular axes ( s = 90° in expression (2.3.2)) when one can invoke the transformation for doubled orientation angles qy of the unit vectors ej and r ... 

On summing this expression over all pairs of neighboring molecules on symmetric two-dimensional lattices, the second term gives no contribution to the sum, and the third one causes only a negligible correction. The main contributions are provided by the last two terms which together make up the structure of the dipole Hamiltonian, with the constants determined by the parameters of the interactions considered. 

In an effort to understand the mechanisms involved in formation of complex orientational structures of adsorbed molecules and to describe orientational, vibrational, and electronic excitations in systems of this kind, a new approach to solid surface theory has been developed which treats the properties of two-dimensional dipole systems.61,109,121 In adsorbed layers, dipole forces are the main contributors to lateral interactions both of dynamic dipole moments of vibrational or electronic molecular excitations and of static dipole moments (for polar molecules). In the previous chapter, we demonstrated that all the information on lateral interactions within a system is carried by the Fourier components of the dipole-dipole interaction tensors. In this chapter, we consider basic spectral parameters for two-dimensional lattice systems in which the unit cells contain several inequivalent molecules. As seen from Sec. 2.1, such structures are intrinsic in many systems of adsorbed molecules. For the Fourier components in question, the lattice-sublattice relations will be derived which enable, in particular, various parameters of orientational structures on a complex lattice to be expressed in terms of known characteristics of its Bravais sublattices. In the framework of such a treatment, the ground state of the system concerned as well as the infrared-active spectral frequencies of valence dipole vibrations will be elucidated. 

Here we focus on the effect of dipolar dispersion laws for high-frequency collective vibrations on the shift and width of their spectral line, with surface molecules inclined at an arbitrary angle 6 to the surface-normal direction. For definiteness, we consider the case of a triangular lattice and the ferroelectric ordering of dipole moments inherent in this lattice type.56,109 Lateral interactions of dynamic dipole moments p = pe (e = (sin os, sin6fcin , cos )) corresponding to collective vibrations on a simple two-dimensional lattice of adsorbed molecules cause these vibrations to collectivize in accordance with the dispersion law 121... 

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