Macroscopic lattice structure examples

Fig. 7.55. At a microscopic scale, roughness is associated with lattice steps, vacancies, and so only which are determined by the lattice structure of the smface. At a macroscopic level, crystallographic character may be revealed in the topographic features, for example, hillocks formed on the (100) surface. It has been found that the etched surface in 25% KOH has well-defined terraces and step features, whereas it has a nodule type of appearance in 50% KOH solution. Table 7.5 provides a summary of the characteristics of the surface etched features in KOH with respect to the crystal orientation of the surfaces.

Other parameters, including the lattice sound speed Cs and weight factor fj, are lattice structure dependent. For example, for a typical D2Q9 (two dimensions and nine lattice velocities see Fig. 1) lattice structure, we have tQ = 4/9, ii 4 = 1/9, f5 8 = 1/36, and = A /3Afi, where Ax is the spatial distance between two nearest lattice nodes. Through the Chapman-Enskog expansion, one can recover the macroscopic continuity and momentum (Navier-Stokes) equations from the above-defined LBM dynamics ... 

Percolation theory rationalizes sizes and distribution of connected black and white domains and the effects of cluster formation on macroscopic properties, for example, electric conductivity of a random composite or diffusion coefficient of a porous rock. A percolation cluster is defined by a set of connected sites of one color (e.g., white ) surrounded by percolation sites of the complementary color (i.e., black ). If p is sufficiently small, the size of any connected cluster is likely to be small compared to the size of the sample. There will be no continuously connected path between the opposite faces of the sample. On the other hand, the network should be entirely connected if p is close to 1. Therefore, at some well-defined intermediate value of p, the percolation threshold, pc, a transition occurs in the topological structure of the percolation network that transforms it from a system of disconnected white clusters to a macroscopically connected system. In an infinite lattice, the site percolation threshold is the smallest occupation probability p of sites, at which an infinite cluster of white sites emerges. 

This review is structured as follows. In the next section we present the theory for adsorbates that remain in quasi-equilibrium throughout the desorption process, in which case a few macroscopic variables, namely the partial coverages 0, and their rate equations are needed. We introduce the lattice gas model and discuss results ranging from non-interacting adsorbates to systems with multiple interactions, treated essentially exactly with the transfer matrix method, in Sec. II. Examples of the accuracy possible in the modehng of experimental data using this theory, from our own work, are presented for such diverse systems as multilayers of alkali metals on metals, competitive desorption of tellurium from tungsten, and dissociative... 

Lorentz-Invariance on a Lattice One of the most obvious shortcomings of a CA-based microphysics has to do with the lack of conventional symmetries. A lattice, by definition, has preferred directions and so is structurally anisotropic. How can we hope to generate symmetries where none fundamentally exist A strong hint comes from our discussion of lattice gases in chapter 9, where we saw that symmetries that do not exist on the microscopic lattice level often emerge on the macroscopic dyneimical level. For example, an appropriate set of microscopic LG rules can spawn circular wavefronts on anisotropic lattices. 

Many of the diamondoids can be brought to macroscopic crystalline forms with some special properties. For example, in its crystalline lattice, the pyramidal-shaped [l(2,3)4]pentamantane (see Table I) has a large void in comparison to similar crystals. Although it has a diamond-like macroscopic structure, it possesses the weak, noncovalent, intermolecular van der Waals... 

In the case of metal clusters, for example, valence electrons show the shell structure which is characteristic of the system consisting of a finite number of fermions confined in a spherical potential well [2]. This electronic shell structure, in turn, motivated some theorists to study clusters as atomlike building blocks of materials [3]. The electronic structure of the metallofullerenes La C60 [4] and K C60 [5] was investigated from this viewpoint. This theorists dream of using clusters as atomlike building blocks was first realized by the macroscopic production of C60 and simultaneous discovery of crystalline solid C60, where C60 fullerenes form a close-packed crystalline lattice [6]. 

Both Bravais lattices and the real crystals which are built up on them exhibit various kinds of symmetry. A body or structure is said to be symmetrical when its component parts are arranged in such balance, so to speak, that certain operations can be performed on the body which will bring it into coincidence with itself. These are termed symmetry operations. For example, if a body is symmetrical with respect to a plane passing through it, then reflection of either half of the body in the plane as in a mirror will produce a body coinciding with the other half. Thus a cube has several planes of symmetry, one of which is shown in Fig. 2-6(a). There are in all four macroscopic symmetry operations or elements reflection. 

Quasi-crystals have macroscopic symmetries which are incompatible with a crystal lattice (Section 2.4.1). The first example was discovered in 1984 when the alloy AlMn is rapidly quenched, it forms quasi-crystals of icosahedral symmetry (Section 2.5.6). It is generally accepted that the structure of quasicrystals is derived from aperiodic space filling by several types of unit cell rather than one unique cell. In two-dimensional space, the best-known example is that of Penrose tiling. It is made up of two types of rhombus and has fivefold symmetry. We assume that the icosahedral structure of AlMn is derived from a three-dimensional stacking analogous to Penrose tiling. As is the case for incommensurate crystals, quasi-crystals can be described by perfectly periodic lattices in spaces of dimension higher than three in the case of AlMn, we require six-dimensional space. 

As examples of the influence of size-induced effects, one can mention the films of refractory metals Mo and W, which in bulk have the bcc structure and at small sizes of 50-100 A have the fee phase [25]. Electronographic methods show that the DS of rare-earth metals Y, Gd, Tb, Ho, and Tm have an fee lattice instead of the hexagonal close-packed lattice (as is the case for macroscopic crystals [15]). 

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