Lattices, simple square

The reciprocal lattices shown in figure B 1.21.3 and figure B 1.21.4 correspond directly to the diffraction patterns observed in FEED experiments each reciprocal-lattice vector produces one and only one diffraction spot on the FEED display. It is very convenient that the hemispherical geometry of the typical FEED screen images the reciprocal lattice without distortion for instance, for the square lattice one observes a simple square array of spots on the FEED display. 

For an fee lattice a particularly simple surface structure is obtained by cutting the lattice parallel to the sides of a cube that forms a unit cell (see Fig. 4.6a). The resulting surface plane is perpendicular to the vector (1,0,0) so this is called a (100) surface, and one speaks of Ag(100), Au(100), etc., surfaces, and (100) is called the Miller index. Obviously, (100), (010), (001) surfaces have the same structure, a simple square lattice (see Fig. 4.7a), whose lattice constant is a/ /2. Adsorption of particles often takes place at particular surface sites, and some of them are indicated in the figure The position on top of a lattice site is the atop position, fourfold hollow sites are in the center between the surface atoms, and bridge sites (or twofold hollow sites) are in the center of a line joining two neighboring surface atoms. 

Figure 9.38 The simplest types of regular 2D lattices (A) the Bethe lattice (Z=3) (B) the honeycomb lattice (Z=3), The simple square lattice (Z=4), (C) the simple square lattice (Z=4), (D) Kagome lattice (Z=4), (E) the triangular lattice (Z=6).

Continuing with our survey of the seven crystal systems, we see that the tetragonal crystal system is similar to the cubic system in that all the interaxial angles are 90°. However, the cell height, characterized by the lattice parameter, c, is not equal to the base, which is square (a = b). There are two types of tetragonal space lattices simple tetragonal, with atoms only at the comers of the unit cell, and body-centered tetragonal, with an additional atom at the center of the unit cell. 

In this subsection we describe a discrete model for vacancy-mediated diffusion of embedded atoms, solve it numerically for the case of In/Cu(0 0 1), and present the results. Our model is defined on a two-dimensional simple square lattice of size / x / (typically, l = 401) centered around the origin. This corresponds to the top layer of a terrace of the Cu(00 1) surface, with borders representing steps. The role of steps in the creation/annihilation of vacancies will be discussed in more detail in the next section. All sites but two are occupied by substrate atoms. At zero time the two remaining sites are the impurity (or tracer) atom, located at the origin, and a vacancy at position (1,0). This corresponds to the situation immediately after the impurity atom has changed places with the vacancy. 

Single-spaced, simple-square lattice of chemisorbed oxygen in (100) azimuth at 27 volts. Multiply current scale by 10... 

Another example is that of lattice chain models. Simple square lattice models were established by Flory as a vehicle for calculating configurational entropies etc., and used later in the simulations of the qualitative behaviour, e.g. of block copolymer phase separation. More sophisticated models such as the bond fluctuation model, and the face centred cubic lattice chain modeP ... 

Figure 2.3(b) The transformation from a simple square lattice to a centred square lattice through the propagation of a linear defect. 

Fio. 8. Diffraction beam intensities as a function of oxygen exposure obtained after a small anneal of the crystal subsequent to ion-bombardment cleaning. Curve 1 Typical beam, in the (110) azimuth at about 28 volts, from the clean nickel lattice. Multiply the ordinate scale by 2 to obtain intensity. Curve 2 Typical beam, in the (001) azimuth at about 58 volts, from the clean nickel lattice. Multiply the ordinate scale hy 6. Curve 3 Typical beam, in the (110) azimuth at about 17 volts, from a double-spaced, face-centered lattice. Curve 4 Typical beam, in the (001) azimuth at about 27 volts, from a single-spaced, simple-square lattice. Multiply the ordinate scale by 2. Curve 5 Typical beam, in the (110) azimuth at about 22 volts, from a nickel oxide lattice. [From Farnsworth and Madden (27).]... 

Fio. 10. Curve 1 Work function versus logjo exposure. Curve 2 Intensity of 27-volt beam from single-spaced, simpled-square structure as a function of logi, exposure. Multiply the ordinate scale by 4. Curve 3 Intensity of 22-volt beam from a nickel-oxide lattice as a function of logjo exposure. [From Farnsworth and Madden (27).]... 

X and Y perpendicular distances. Bloch, Weisman, and Varma were the first to point out the significance of disorder for the transport properties of such materials, introducing the so-called "disorder model" and interpreting the temperature-dependence of 6 in terms of hopping conduction. Consider as an example a simple square lattice of molecular stacks withoC = 0 = and X +Y =S, Let S be the minimum distance... 

Short homopolymeric chains restricted to a simple square lattice and to a simple cubic lattice have been studied by Chan and Dill [59-61]. For very small systems, all compact conformations could be enumerated. Not surprisingly, significant intrachain entropy loss occurs on collapse. Expand-... 

In order to get more experience with the newly proposed index ( ) we will consider the leading eigenvalue X of D/D matrices for several well-defined mathematical curves. We should emphasize that this approach is neither restricted to curves (chains) embedded on regular lattices, nor restricted to lattices on a plane. However, the examples that we will consider correspond to mathematical curves embedded on the simple square lattice associated with the Cartesian coordinates system in the plane, or a trigonal lattice. The selected curves show visibly distinct spatial properties. Some of the curves considered apparently are more and more folded as they grow. They illustrate the self-similarity that characterizes fractals. " A small portion of such curve has the appearance of the same curve in an earlier stage of the evolution. For illustration, we selected the Koch curve, the Hubert curve, the Sierpinski curve and a portion of another Sierpinski curve, and the Dragon curve. These are compared to a spiral, a double spiral, and a worm-curve. 

Baldock computed the bond orders for several kinds of lattices formed by alkaline-like atoms planar square lattice, simple cubic lattice, bcc lattice He showed that the bond orders are much... 

FIGURE 5.6. A trapped conformation for the algorithm with only local moves for a chain on the simple square lattice (A). A longer distance move that guarantees the ergodicity of the algorithm, where a U shaped fragment at one part of the chain is cut-off and attached somewhere else (B). 

In our above simple square lattice, we have considered only nearest neighbors and the bonding between them. Let us consider now the case where we also include occupied second nearest neighbors [11]. In that case, under the same site occupation probability p, where farther (second nearest) neighbors are also considered, the percolation onset will be achieved at a p for which no percolation existed in the nearest neighbors-only case. Let us define now the critical pc values for the two... 

Let us consider the monolayer adsorbed film formed on a heterogeneous surface in equilibrium with a gas of the chemical potential (ji). In the simplest version of the lattice model, every site i is characterized by the occupation variable, which equals 1 when the site is occupied by the gas atom and equals 0 when it is empty. A simple square lattice is assumed. In this case, the Hanultonian for the adsorbed monolayer can be written as [234]... 

More complex expressions for the gas-solid potential are found when the dependence on the 2D distance (parallel to the surface) is introduced. Thus, the model given by Hill [173] is based on a simple square lattice, the potential being a separable function of the planar 2D coordinates and the coordinate perpendicular to the surface, and the periodic component of the energy being described by a simple sinusoidal function which is independent of the perpendicular distance. Doll and Steele [18] have indicated, nevertheless, that Hill s model is oversimplified. For example, it considers a simple square lattice surface when it is the close-packed array, which is the most interesting case from an experimental point of view. Obviously, for heterogeneous surfaces, a more complex dependence on x, y, and z must be considered [6,11]. 

A very simple lattice model that could be treated exactly by full enumeration of all compact states in short chains is the HP model, extensively studied by Dill et al." The polypeptide is represented as a string of hydrophobic (H) and hydrophilic (P) beads (residues) on a simple square or cubic lattice. Hydrophobic residues attract each other, while the remaining possible pairwise interactions are equal to zero. 

Layered Planar Hexagonal and Simple Square Lattices. 235... 

For a 2D square lattice q = 4, and the high- and low-temperature expansions are related in a simple way... 

A simple illustrative example of reciprocal space is that of a 2D square lattice where the vectors a and b are orthogonal and of length equal to the lattice spacing, a. Here a and b are directed along the same directions as a and b respectively and have a length 1/a... 

To present briefly the different possible scenarios for the growth of multilayer films on a homogeneous surface, it is very convenient to use a simple lattice gas model language [168]. Assuming that the surface is a two-dimensional square lattice of sites and that also the entire space above the surface is divided into small elements, forming a cubic lattice such that each of the cells can be occupied by one adsorbate particle at the most, the Hamiltonian of the system can be written as [168,169]... 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

The easiest ciystal lattice to visualize is the simple cubic stracture. In a simple cubic crystal, layers of atoms stack one directly above another, so that all atoms lie along straight lines at right angles, as Figure 11-26 shows. Each atom in this structure touches six other atoms four within the same plane, one above the plane, and one below the plane. Within one layer of the crystal, any set of four atoms forms a square. Adding four atoms directly above or below the first four forms a cube, for which the lattice is named. The unit cell of the simple cubic lattice, shown in... 

Yet another common crystal lattice based on the simple cubic arrangement is known as the face-centered cubic structure. When four atoms form a square, there is open space at the center of the square. A fifth atom can fit into this space by moving the other four atoms away from one another. Stacking together two of these five-atom sets creates a cube. When we do this, additional atoms can be placed in the centers of the four faces along the sides of the cube, as Figure 11-28 shows. 

---