Comer atoms

Body-centered cubic cell (BCC). This is a cube with atoms at each comer and one in the center of the cube. Here again, comer atoms do not touch each other. Instead, contact occurs along the body diagonal the atom at the center of the cube touches atoms at opposite comers. 

Although the comer atoms must move apart to convert a simple cube into a body-centered cube, the extra atom in the center of the stracture makes the body-centered cubic lattice more compact than the simple cubic structure. All the alkali metals, as well as iron and the transition metals from Groups 5 and 6, form ciystals with body-centered cubic structures. 

A In a bcc unit cell, there are eight comer atoms, of which — of each is apportioned to the unit... 

In a fee unit cell the number of atoms is computed as 1/8 atom for each of the eight comer atoms (since each is shared among eight unit cells) plus 1/2 atom for each of the six face atoms (since each is shared between two unit cells). This gives the total number of atoms per unit cell as atoms/unit cell = (1/8 comer atom x 8 comer atoms/unit cell) + (1/2 face atom x 6 face atoms/unit cell) = 4 atoms/unit cell... 

Keep in mind that for close-packed structures, the atoms touch each other in all directions, and all nearest neighbors are equivalent. Let us first examine the HCP structure. Figure 1.17 is a section of the HCP lattice, from which you should be able to see both hexagons formed at the top and bottom of what is called the unit cell. You should also be able to identify the ABA layered structure in the HCP unit cell of Figure 1.17 through comparison with Figure 1.16. Let us count the number of atoms in the HCP unit cell. The three atoms in the center of the cell are completely enclosed. The atoms on the faces, however, are shared with adjacent cells in the lattice, which extends to infinity. The center atoms on each face are shared with one other HCP unit cell, either above (for the top face) or below (for the bottom face), so they contribute only half of an atom each to the HCP unit cell under consideration. This leaves the six corner atoms on each face (12 total) unaccounted for. These comer atoms are at the intersection of a total of six HCP unit cells (you should convince yourself of this ), so each comer atom contributes only one-sixth of an atom to our isolated HCP unit cell. So, the total number of whole atoms in the HCP unit cell is... 

In a manner similar to that nsed to calculate the density of a nnit cell, we can calcn-late the density of atoms on a plane, or planar density. The perpendicular intersection of a plane and sphere is a circle, so the radius of the atoms will be helpful in calcnlat-ing the area they occnpy on the plane. Refer back to Example Problem 1.4 when we calcnlated the lattice parameter for a BCC metal. The section shown along the body diagonal is actually the (110) plane. The body-centered atom is entirely enclosed by this plane, and the comer atoms are located at the confluence of four adjacent planes, so each contributes 1/4 of an atom to the (110) plane. So, there are a total of two atoms on the (110) plane. If we know the lattice parameter or atomic radius, we can calculate the area of the plane, Ap, the area occupied by the atoms, Ac, and the corresponding... 

Fig. 38. Left, bisphenoid of PbCla. Centre and right, arrangements of atoms on 111 and 111. The atoms depicted are those which lie on, or not far below, the plane of the comer atoms.

The compound four-atom unit cell of copper is termed face-centred the cubic unit cell has atoms not only at the corners but also at the centre of each face. If the various planes arc examined in the same way as for a-iron, it will be seen that in the first place 010 is absent, because the 010 planes (Fig. 125) comprising one comer atom and one of the f ace-... 

Initial work indicates that dispersed metals may be used to promote a variety of organometallic reactions. The Heck Arylation proceeds smoothly over supported Pd catalysts while diene cyclizations can be catalyzed by dispersed Rh metal. The use of these heterogeneous species facilitates product isolation and permits the application of flow systems rather than batch reactors for these reactions. Frontier Molecular Orbital and mechanistic considerations indicate that these reactions take place on the coordinately unsaturated comer atoms on the metal surface. 

We have used our Single Turnover (STO) reaction sequence to characterize dispersed metal catalysts with respect to the numbers of alkene saturation sites, double bond isomerization sites, and hydrogenation inactive sites they have present on their surfaces (ref. 13). Comparison of the product composition observed when a series of STO characterized Pt catalysts were used for cyclohexane dehydrogenation with those observed using a number of instrumentally characterized Pt single crystal catalysts has shown that the STO saturation sites are comer atoms of one type or another on the metal surface (ref. 10). 

Fig. 2. Energy levels of the 5s, 5p, and 4d electrons of a Pd octahedral comer atom...

When a series of STO characterized Pd/A Og catalysts were used to promote the Heck reaction (Eqn. 1) the amount of the fi aryl enol ethers, 1 and 2, formed after a 60 minute reaction was directly related to the comer site densities on these catalysts. Thus, this reaction and presumably, others such as the diene cyclization shown in Eqn. 2, which require the adsorption of two reactive species on a single surface atom, must take place on the more coordinatively unsaturated comer atoms. 

The upper drawing in Fig. 34 is a schematic, electron-domain representation of the spin-density in a plane through two neighboring comer atoms and the adjacent center atom in Slater s model of the alkali metals. Solid circles represent the atoms kernels (M+ cations). The Pauli Exclusion Principle permits domains occupied by electrons of opposite spin to overlap (comer atoms with the central atom), but prohibits overlap between domains occupied by electrons of the same spin (comer atoms with comer atoms). 

Differences in Pt-H bond strength may result from several reasons. First, the metal clusters in a heterogeneous catalyst are small and therefore contain only small crystal surfaces and many edges and corners. The atoms on the corners and edges are coordinatively highly unsaturated and therefore will bond strongly to H atoms. So the strongest bonded H atoms may be the H atoms that are chemisorbed on the corners of the clusters. After saturation of the cluster with a monolayer of H, these comer-atoms will still be relatively unsaturated and can bond to yet another H atom. This H however will be bonded rather weakly. 

Figure 5. The 55-atom cubooctahedron as a model for the most abundant structure in EUROPT-1. (1) cyclopentane adsorbed on a comer atom, (2) a Cs cyclic species on the (100) surface (3) a C6 species en route to benzene on the (111) surface [7]...

Figure 3.9. Simplified scfiematic of tfie transformation from BCC to FCC, exfiibited between the three allotropes of iron. Comer atoms have been omitted for clarity.

The diagram below shows a space-filling diagram of a face-centered cubic unit cell cut to show only the part of each atom that is inside the unit cell boundaries. The comer atoms are each shared among eight unit cells, so of the atom is in the unit cell shown. The face-centered atoms are shared between two unit cells, so 5 of the atom is in the unit cell shown. The eight corners of the unit cell then total 8 X g = 1 atom, the six faces total 6 X j = 3 atoms, and there is a total of 4 atoms in the unit cell. 

If another sphere is added in the center of the simple cubic stracture, the result is called body-centered cubic (bcc). If the added sphere has the same radius as the others, the size of the unit cell expands so that the diagonal distance through the cube is 4r, where r is the radius of the spheres. The comer atoms are no longer in contact with each other. The new unit cell is 2.3 Ir on each side and contains two atoms because the body-centered atom is completely within the unit cell. This cell has two lattice points, at the origin (0, 0, 0) and at the center of the cell... 

Show that a sphere of radius 0.73r, where r is the radius of the comer atoms, will fit in the center of a primitive cubic structure. 

Simulations of self-diffusion have been reviewed recently [60]. In addition to molecular motion on flat surfaces (including those with atomic roughness), selfdiffusion constants have been evaluated for atoms adsorbed on surfaces with comers (as in pores with rectangular cross sections or on grooved surfaces) and with steps. In these systems, a deep nearly one-dimensional potential well occurs in the model gas-surface energy at the comers. Atoms adsorbed in this well are essentially localized in one-dimension, which means that self-diffusion hardly occurs in the directions perpendicular to the comer. 

III) on faces. With the ultradispersed catalysts, the particles were thought to be two-dimensional rafts composed of about eight rhodium atoms. Such particles would have only comer atoms. With somewhat larger three-dimensional particles the number of edge atoms would increase. With larger particles there would be an increase in the number of face atoms. A more complete discussion of the nature of the active sites on a catalyst surface is presented in Chapter 3. 

As discussed previously, this reaction was also run under these same conditions over the series of specifically cleaved platinum single erystals shown in Fig. 3.2. 3 The results of these experiments show that it was the corner atoms on these crystals that promoted C-H bond breaking. Thus, the saturation sites on the dispersed metal catalysts are also comer atoms. Since this saturation site description agrees with that proposed on the basis of the butene deuteration described previously,5 -62 it is likely that the isomerization sites, M, are edge atoms and the hydrogenation inactive sites, M, are face atoms. A similar approach can be used to determine the nature of the active sites responsible for promoting almost any type of reaction. 5.70... 

The 111 plane can intersect another plane at either 60 or 120 so there are two 111-111 edge sites, labeled D and E, and two 111-100 edge sites, labeled C and F. The 90 orientation required for the intersection of two 100 planes is not possible. The seven different comer atoms range from the cubooctahedral, F, with six neighboring atoms to the tetrahedral comer, M, having only three. Here, too, the 60 and 120 plane intersections show up in the comer atom arrangements. Stmetures of these sites are depicted in Fig. 4.5.7,8... 

Other views of some of these sites are illustrated with the crystal shapes shown in Figs. 4.4 and 4.6. It can be seen in the cubic crystal that the atoms on the comer are not bonded along the edges of the cube. Instead these comer atoms are bonded to three nearest neighbors and are, in reality, tetrahedral comers (M) as shown more clearly on the tetrahedral crystal. The face atoms on the cube are in a 100 plane (B). What appear to be edge atoms, though, are the surface sites shown in Fig. 4.5 as the 111-111-100-100 comer sites (H). 

If the metal catalyst particles were present only in the form of these idealized crystals, then the number of active comer atoms present would be very low. However, STO evaluations of dispersed metal catalysts have shown that these active atoms are present in rather large amounts, at times as high as 30%-35% of the total metal atoms present. Such high surface concentrations of the highly unsaturated atoms can only be accounted for by the presence of the irregular particle shapes that were observed using dark field TEM imaging techniques. Additional active sites are probably present as adatoms on the 111 (M) and 100 (K) planes as shown in Fig. 4.4. 

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