Corner atoms

To calculate the fraction of occupied space in a close-packed structure, we considei a ccp structure, e can use the radius of the atoms to find the volume of the cube and ow muc o t at volume is taken up by atoms. First, we look at how the cube is built rom t e atoms. In Fig. 5.29, we see that the corners of the cubes are at the centers of etg t atoms, n y 1/8 of each corner atom projects into the cube, so the corner atoms collectively contribute 8xi/S=1 atom to the cube. There is half an atom on each of t e six aces, so the atoms on each face contribute 6 X 1/2 = 3 atoms, giving four... 

The number of atoms in a unit cell is counted by noting how they are shared between neighboring cells. For example, an atom at the center of a cell belongs entirely to that cell, but one on a face is shared between two cells and counts as one-half an atom. As noted earlier for an fee structure, the eight corner atoms contribute 8 X 1/8 = 1 atom to the cell. The six atoms at the centers of faces contribute 6x1/2 = 3 atoms (Fig. 5.37). The total number of atoms in an fee unit cell is therefore 1 + 3=4, and the mass of the unit cell is four times the mass of one atom. For a bcc unit cell (like that in Fig. 5.34b), we count 1 for the atom at the center and 1/8 for each of the eight atoms at the vertices, giving 1 + (8 X 1/8) = 2 overall. 

The increase in the rate of case If is related with an increase in the relative ratio of the edge and corner atoms over the decreasing number of terrace atoms. This increase in reactivity relates to the increased degree of coordinative unsaturation of the edge and corner atoms. 

Table 1.3 Methane activation on edge and corner atoms (kilojoules per mol).

We discussed that for methane activation this leads to lowering of the activation energy compared to the reactivity of terrace, edge, or corner atoms successively. 

As can be seen from Figure 1.20 [22], those transition states that do not share binding to the same surface metal atom have low barriers. The fcc(lOO) surface has the unique property that the reaction can occur through motion over the square hollow with bonds that remain directed toward the corner atoms of the square atom arrangement on the surface. This is a unique and important feature of reactions that require in their transition states interactions with several surface atoms. 

Basically, zeolites consist of Si04 and AIO4 tetrahedra (Fig. 5.28), which can be arranged by sharing 0-corner atoms in many different ways to build a crystalline lattice (Fig. 5.29). 

Edge atoms predominate in the region around dK 5, and only at still smaller diameters does the fraction of corner atoms become very large. 

Fig. 16.10. Topograph and local spectroscopy of Si(lll) with oxygen. Left, the STM image of a region of the oxygen-exposed Si( 111 )-7 X 7 surface. Right, the local tunneling spectra at different sites. Spectra A, B, and C are those of unreacted restatom, corner atom, and center adatom, respectively. Spectra D, E, and F are obtained over oxygen-induced dark, bright, and perturbed (gray) adatom sites, respectively. (Reproduced from Avouris, Lyo, and Bozso, 1991, with permission.)...

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 alloys with 0-23% Cu the activation energy of the total conversion of n-hexane is only marginally influenced and the observed effects are consequently connected with the preexponential factors. Since the selectivity of nickel diluted with copper is near the value found by Anderson et al. (113) for highly dispersed films, considering a common cause is suggested (60). Anderson assumes that with a large fraction of surface atoms in very small crystals the isolated corner atoms favor the formation of carbocyclic intermediates of isomerization, whereas hydrogenolysis requires two or more adjacent platinum atoms in a crystal plane. 

Carbon atoms bonded to four other atoms are each at the center of a tetrahedron. As shown below for ethane, H3C-CH3, the two tetrahedrons are joined so that the central carbon atom of one is a corner atom of the other. 

A body-centered cubic unit cell has eight corner atoms plus an additional atom in the center of the cube (Figure 10.22b). This body-centered cubic unit cell, with two repeating offset layers and with the spheres in a given layer slightly separated, is the repeat unit found in body-centered cubic packing. 

As shown in Figure 10.22a, there is an atom at each of the eight corners of the primitive-cubic unit cell. When unit cells are stacked together, each corner atom is shared by eight cubes, so that only 1 / 8 of each atom "belongs" to a given unit cell. Thus there is 1/8 X 8 = 1 atom per unit cell. 

Cubic closest-packing uses a face-centered cubic unit cell. Looking at any one face of the cube head-on shows that the face atoms touch the corner atoms along the diagonal... 

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