Unit mesh

Figure 1 Unit meshes and 20 Miiier indices, (a) Exampies of 20 Bravais nets (b) exam-pies of famiiies of rows with Miiier indices referenced to the unit mesh vectors. (11) and (31) famiiies of rows are shown.

Figure 2 Examples of overlayer structures with appropriate notation, (a) fee (100) p(2X2) (b) fee (100) c(2X2) (c) fee (111) p(73 X 73>R30< (d) bee (110 ) p(2 X 1) and bee (110) p(1 X 2). The two orientations for the unit mesh in (d) are both possible because of the symmetry of the substrate they have the same free energy and are called degenerate.

Any periodicity in the surfiice or overlayer different from that of the clean unreconstructed surface produces addidonal diffracted beams. A unit mesh latter than... 

The simplest diffraction measurement is the determination of the surface or overlayer unit mesh size and shape. This can be performed by inspection of the diffraction pattern at any energy of the incident beam (see Figure 4). The determination is simplest if the electron beam is incident normal to the surface, because the symmetry of the pattern is then preserved. The diffraction pattern determines only the size and shape of the unit mesh. The positions of atoms in the surface cannot be determined from visual inspection of the diffraction pattern, but must be obtained from an analysis of the intensities of the diffracted beams. Generally, the intensity in a diffracted beam is measured as a fimction of the incident-beam energy at several diffraction geometries. These intensity-versus-energy curves are then compared to model calculations. ... 

Subsequently, Mitchell s group in Vancouver, by means of a tensor-LEED study17 of the Cu (110)-(2 x 3)N surface structure, supported a reconstruction model in which the topmost layer is described as a pseudo-(100)-c(2 x 2)N overlayer with metal corrugation of about 0.52 A in the reconstructed layer. Each nitrogen adatom is almost coplanar with the local plane formed by the four neighbouring copper atoms. Of the four N atoms present in the unit mesh, three are also bonded to Cu atoms in the layer below and therefore are five coordinate. 

Figure 2. Schematic representation of the Silicalite-I unit mesh. Its developed inner surface has also been represented to emphasize its relative complexity.

The surface structures observed for the multilayer phthalocyanine films are summarized in Table 5.7. These structures do not correspond to planes of either of the previously reported crystal structures of vapor deposited phthalocyanine films since the unit mesh constants reported in this work are considerably smaller than those previously reported. The unit cell dimensions correspond much more closely to a unit cell containing one molecule rather than, for example, the four molecules per cell reported for a-phthalocyanine. 

Figure 1.2. Plan view of the Ni(lll)c(4 x 2)-CO surface phase showing the bridge site model favoured for many years on the basis of the interpretation of vibrational spectroscopy, and the mixed-hollow site model subsequently established through SEXAFS, PhD, and LEED measurements. To allow visibility of all surface atoms the C atoms are shown as the larger dark-shaded spheres and the atoms as smaller white spheres. The full lines show the primitive unit mesh while the dashed lines show the centred (4 x 2) unit mesh.

Figure 1.3. Plan view of the Ni(100)c(2 x 2)-0 and Ni(100)(2 x 2)-C p4g surface structures. In each case the full lines show the primitive unit mesh while in the -induced structure the dashed lines show the centred (2 x 2) mesh. In the case of the C-induced structure the outermost Ni atoms are shown as smaller more-darkly shaded spheres than those of the underlying substrate to see more clearly the relationship of this reconstructed layer to the substrate. Notice that this reconstruction also leads to some reduced Ni-Ni nearest-neighbour distances in the surface, so using the bulk atomic radii for these atoms would lead to some overlapping spheres.

Figure 1.4. Plan view of Cu(l 00)(x 2 /2)R45°-0 surface reconstruction. The outermost layer Cu atoms are shown more lightly shaded than those of the underlying substrate to show more clearly the missing-row structure of this outermost layer. The full lines show the surface unit mesh.

Figure 1.6. Plan and side views of the structure of the Cu(410)-O surface phase. The full and dashed lines show respectively the primitive and centred rectangular surface unit meshes (that are unchanged from those of the clean surface by the adsorption).

Figure 1.11. Heterochiral ordered structures of glycinate on Cu(l 10) and Cu(100). The full lines show the primitive unit mesh in each case, while dashed lines show the location of glide symmetry lines.

A crystal is an orderly array of atoms or molecules but, rather than focusing attention on these material units, it is helpful to consider some geometrical constructs that characterize its structure. It is possible to describe the geometry of a crystal in terms of what is called a unit cell a parallelepiped of some characteristic shape that generates the crystal structure when a three-dimensional array of these cells is considered. We then speak of the lattice defined by the intersections of the unit cells on translation through space. Since we are interested in crystal surfaces, we need to consider only the two-dimensional faces of these solids. In two dimensions the equivalent of a unit cell is called a unit mesh, and a net is the two-dimensional equivalent of a lattice. Only four different two-dimensional unit meshes are possible. 

FIG. 9.17 Illustrations of LEED spots and coherent structures formed by adsorbed species, (a) indexing of the LEED spots of W(100) pattern in Figure 9.16a as described in Example 9.7 (b) side view of coherent structures formed by adsorbed (open circles) species on metal surface (c) unit mesh for W(100) with adsorbed oxygen (open circles) (d) unit mesh for W(100) with adsorbed hydrogen (open circles). 

In the adsorption studies we have discussed, the expansion of the unit mesh is the same in both directions, but this need not be the case. Examples in which the expansion along different axes of the mesh varies are p(4 x 2)—O for the adsorption of 02 on Mo(l 11), p(3 x 15)—O for 02 on Pt(l 11), c(4 x 2)—S for H2S on Au(100), and c(9 x 5)—CO for CO on W(110). Somorjai (1981, 1994) has assembled extensive tables of this sort of information. Note that many but not all adsorbates are dissociated on the metal surfaces. Finally, it is not necessary for the supernet and the substrate to show the same angles between sides of their respective meshes. The Wood notation does not apply in these cases, but an alternative notation exists... 

Solution Use the law of cosines to determine the length of the two diagonals of the substrate unit mesh ... 

Firment and Somorjait showed that the C4-C8 /7-alkanes adsorb on the Pt(lll) surface in ordered monolayers for which the unit mesh is a parallelogram with the following dimensions ... 

Look up the formula for the area of a parallelogram and calculate the area per mole, assuming one molecule per unit mesh. Compare this area with the Pt(lll) unit mesh for which the dimensions are also given. Do the data make any more sense if it is assumed that some of these alkanes form surface structures with two molecules per unit mesh Prepare a plot of the area per molecule versus the number of carbon atoms in the chain. Criticize or defend the following proposition The amount of free area per unit mesh in these packings is equivalent to one Pt mesh. 

We call this Pt(100) surface reconstructed. Surface reconstruction is defined as the state of the clean surface when its LEED pattern indicates the presence of a surface unit mesh different from the bulklike (1 x 1) unit mesh that is expected from the projection of the bulk X-ray unit cell. Conversely, an unreconstructed surface has a surface structure and a so-called (1 x 1) diffraction pattern that is expected from the projection of the X-ray unit cell for that particular surface. Such a definition of surface reconstruction does not tell us anything about possible changes in the interlayer distances between the first and the second layers of atoms at the surface. Contraction or expansion in the direction perpendicular to the surface can take place without changing the (1 x 1) two-dimensional surface unit cell size or orientation. Indeed, several low Miller index surfaces of clean monatomic and diatomic solids exhibit unreconstructed surfaces, but the surface structure also exhibits contraction or expansion perpendicular to the surface plane in the first layer of atoms (9b). 

In two dimensions, the equivalents of unit cell and lattice are unit mesh and net, respectively. Crystallography in two dimensions is, obviously, simpler than that in three dimensions, and there are only five types of net (illustrated in Figure 5.15). The choice of unit mesh is arbitrary. The primitive unit mesh (illustrated at the bottom left hand comer of each net) is the smallest possible repeating quadrilateral with lattice points only at the comers. However, it may be appropriate... 

The structure of a surface layer (be it the surface of a pure solid or a monolayer of adsorbed gas) usually differs from that of the underlying substrate. A shorthand notation exists in which the unit mesh of the surface layer is described in terms of the unit mesh of the layer immediately below it. Some examples are illustrated in Figure 5.17. The prefix C indicates that the unit mesh for the surface layer contains centre atom(s) and R indicates that this unit mesh is rotated by the stated angle with respect to the substrate unit mesh. 

In examples (a) to (c), the locations of the adsorbed atoms differ in that they are (a) end-on, (6) in a two-fold bridging position, and (c) in a three-fold well position in relation to the substrate atoms. However, the outer surface net is the same in each case and, as such, contributes to the same diffraction pattern. In example (d), the alternative unit meshes shown are both correct and both would lead to the same ultimate interpretation, but the C(2 x 2) unit mesh offers the greater convenience. 

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