The Two-dimensional Lattice

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. 

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 unit cell of the p(2x2) adsorbate structure in Fig. 5.7 is twice as large in both directions as the unit cell of the substrate and hence the structure is called p(2x2), where the p stands for primitive. The coverage corresponds to 0.25 monolayers (commonly abbreviated to ML). 

The Wood notation, as this way of describing surface structures is called, is adequate for simple geometries. However, for more complicated structures it fails, and one uses a 2x2 matrix which expresses how the vectors al and a2 of the substrate unit cell transform into those of the overlayer. 

A well-known example of adsorbate induced surface reconstruction is that of carbon on the (100) surface of nickel. Even though this surface already offers a four-fold coordination to the carbon atom in unreconstructed form, additional energy is gained by the so-called clock-anti-dock reconstruction shown in Fig. 5.8. Of course, it costs energy to rearrange the surface nickel atoms, but this investment is more than compensated by 

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Where r represents the distances on the pattern, we must determine constants A,B and y of the two-dimensional lattice (Fig. 8) In real situations, reflections on texture patterns are in the sharp of arcs. This, may be explained as follows. The intersection of the rings of the reciprocal lattice with a plane gives, in the ideal case, a point. However with real textures, because of a certain disorder in the crystal orientation relative to the texture... 

Note Depending on the order in the molecular stacking in the columns and the two-dimensional lattice symmetry of the columnar packing, the columnar mesophases may be classified into three major classes hexagonal, rectangular and oblique (see Definitions 3.2.2.1. to 3.2.2.3). 

Fig. 15. Plan views of the two-dimensional lattice of the columns in columnar rectangular (a) to (c) and oblique (d) mesophases. Ovals indicate the planes of the molecular discs.

The unit cells for the two-dimensional lattices are parallelograms with their corners at equivalent positions in the array (i.e., the corners of a unit cell are lattice points). In Figure 1.17, we show a square array with several different unit cells depicted. All of these, if repeated, would reproduce the array it is conventional to choose the smallest cell that fully represents the symmetry of the structure. Both unit cells (la) and (lb) are the same size but clearly (la) shows that it is a square array, and this would be the conventional choice. Figure 1.18 demonstrates the same principles but for a centred rectangular array, where (a) would be the conventional choice because it includes information on the centring the smaller unit cell (b) loses this information. It is always possible to define a non-centred oblique unit cell, but doing so may lose information about the symmetry of the lattice. 

It is much simpler to calculate an interpolymer interaction which is sufficiently well modelled by adsorption of chains on a one-dimensional lattice, than to calculate adsorption on the two-dimentional one as well as to calculate adsorption of small particles on the macromolecule. When calculating adsorption on the two-dimensional lattice (usual surface), for example, difficulties rapidly pile up with increasing the degree of filling of the surface with macromolecules it is difficult to take into consideration self- and intercrossings of the adsorbed macromolecules, the influence exerted by the length and rigidity of these macromolecules, etc. 

Fig. 8.1. Configurations of the adspecies in the first coordination sphere (c.s.) of the central sites of the two-dimensional lattices with z = 4 and 6.

Eqs. (31) and (32) becomes more complicated. The dimensionality of the set of equations, however, coincides with that of the system in the QCA. A more exact description is obtained with the correlators of greater dimensionality m>2 (see, e.g., Refs. [90,91]). Of special interest are the one-dimensional systems with s — 2. Exact solutions for the migrating adspecies on the one-dimensional lattice have been obtained for a small number of sites [92]. In Refs. [93,94] the procedure of numerical analysis of the hierarchical system of equations has been elaborated, which is applicable not only to the one-dimensional [94,95] but also to the two-dimensional lattices [95,96], as well, the interaction with the second neighbors being taken into account (d — 1) [97]. Also, it should be noted that the expansions (virial or diagrammatic) [98] similar to the common expansion in the equilibrium theory of condensed systems [77] are used for closing the kinetic equations. 

Figure 32. The visualization of dynamic percolation. A set of the realization of the occupied sites in the two-dimensional lattice over time with fixed time increments.

Herein, two neighboring rows of the two-dimensional lattice are labeled by... 

Figure 5.12. The illustration of the two-dimensional lattice with one long (b ) and one short (a ) reciprocal lattice vectors. If the three lowest Bragg angle peaks (filled circles) are selected as a basis set for indexing, all of them are collinear and only depend on a. The remaining two lattice parameters b and y ) cannot be determined from this basis set.

Now we to concentrate on the properties of the two-dimensional lattice, the space-time lattice. The partition function, Eq. (133), shows that there is coupling only in the time direction and only between nearest-neighbor time slices. This allows us to use the statistical mechanics technique of writing the partition function Z of the finite system as the trace of a matrix T to the power Nt. 

For a purely monolayer model, the exact Onsager [82] solution for the two-dimensional lattice problem, predicts the so-called logarithmic discontinuity on the heat capacity curve at the critical temperature T. The deviations of the actual adsorption systems from the ideal Onsager model mean that, instead of a logarithmic discontinuity, more or less rounded peaks are observed, centered at T = Tc. The surface energetic heterogeneity is believed to be the main source of that rounding. 

Critical properties of the two-dimensional lattice gas of Lennard-Jones particles on a square lattice have been studied by Patrykiejew and Borowski [116] with the help of Monte Carlo version of the coherent anomaly method (CAM) developed by Suzuki and coworkers [115], as well as by the conventional Monte Carlo simulation [105]. The detailed presentation of the coherent anomaly method is well beyond the scope of this chapter. Therefore, here I confine myself to a brief description of its foundations and then present the results relevant to the considered problems. 

Three-dimensional lattices use the same nomenclature as the two-dimensional lattice described above. If any lattice point is chosen as the origin, the position of any other lattice point is defined by the vector P(uvw) ... 

As with the two-dimensional lattices, the three-dimensional (Bravais) lattices, the direct lattices, are said to occupy real space, and the reciprocal... 

The reflections of Bragg peaks can be indexed by two Miller indices, hk. Their angular positions, 29i, , yield the lattice plane spacing dfj, = 2nlq for the two-dimensional lattice structure. It is possible to have an estimate of the dimensions of the domains (D ) by analyzing the full width at half maximum (FWHM) of a GIXD peak (A p), corrected from the instrumental broadening with the Scherrer equation ... 

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