Face-Centered Cubic Materials

The simple cubic crystal structure we discussed above is the simplest crystal structure to visualize, but it is of limited practical interest at least for elements in their bulk form because other than polonium no elements exist with this structure. A much more common crystal stmcture in the periodic table is the face-centered-cubic (fee) structure. We can form this structure by filling space with cubes of side length a that have atoms at the corners of each cube and also atoms in the center of each face of each cube. We can define a supercell for an fee material using the same cube of side length a that we used for the simple cubic material and placing atoms at (0,0,0), (0,g/2,g/2), (g/2,0,g/2), and (g/2,g/2,0). You should be able to check this statement for yourself by sketching the structure. 

The primitive cell for the fee metal can be defined by connecting the atom at the origin in the structure defined above with three atoms in the cube faces adjacent to that atom. That is, we define cell vectors 

These vectors define the fee lattice if we place atoms at positions 

The results from calculating the total energy of fee Cu as a function of the lattice parameter are shown in Fig. 2.3. The shape of the curve is similar to the one we saw for Cu in the simple cubic crystal structure (Fig. 2.1), but 

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Figure 4.5 shows another surface that can be defined in a face-centered cubic material this time the highlighted plane intercepts the x, y, and z axes at 1, 1, and 1. The reciprocals of these intercepts are 1/1, 1/1, and 1/1. 

The most important metals for catalysis are those of the groups VIII and I-B of the periodic system. Three crystal structures are important, face-centered cubic (fee Ni, Cu, Rh, Pd, Ag, Ir, Pt, Au), hexagonally dose-packed (hep Co, Ru, Os) and body-centered cubic (bcc Fe). Figure 5.1 shows the unit cell for each of these structures. Note that the unit cells contain 4, 2, and 6 atoms for the fee, bcc, and hep structure, respectively. Many other structures, however, exist when considering more complex materials such as oxides, sulfides etc, which we shall not treat here. Before discussing the surfaces that the metals expose, we mention a few general properties. 

The same atom-centered polyhedra can be used to describe interstitial diffusion in all the many metal structures derived from both face-centered cubic and hexagonal closest packing of atoms. In these cases the polyhedra are centered upon a metal atom and all the tetrahedral and octahedral interstitial sites are empty. The hardening of metals by incorporation of nitrogen or carbon into the surface layers of the material via interstitial diffusion will use these pathways. 

The porous membrane templates described above do exhibit three-dimensionality, but with limited interconnectedness between the discrete tubelike structures. Porous structures with more integrated pore—solid architectures can be designed using templates assembled from discrete solid objects or su-pramolecular structures. One class of such structures are three-dimensionally ordered macroporous (or 3-DOM) solids, which are a class of inverse opal structures. The design of 3-DOM structures is based on the initial formation of a colloidal crystal composed of monodisperse polymer or silica spheres assembled in a close-packed arrangement. The interconnected void spaces of the template, 26 vol % for a face-centered-cubic array, are subsequently infiltrated with the desired material. 

Figure 1.18 The face-centered cubic (FCC) structure showing (a) atoms touching and (b) atoms as small spheres. Reprinted, by permission, from W. Callister, Materials Science and Engineering An Introduction, 5th ed., p. 32. Copyright 2000 by lohn Wiley Sons, Inc.

The transition metal carbides do have a notable drawback relative to engineering applications low ductility at room temperature. Below 1070 K, these materials fail in a brittle manner, while above this temperature they become ductile and deform plastically on multiple slip systems much like fee (face-centered-cubic) metals. This transition from brittle to ductile behavior is analogous to that of bee (body-centered-cubic) metals such as iron, and arises from the combination of the bee metals strongly temperature-dependent yield stress (oy) and relatively temperature-insensitive fracture stress.1 Brittle fracture is promoted below the ductile-to-brittle transition temperature because the stress required to fracture is lower than that required to move dislocations, oy. The opposite is true, however, above the transition temperature. 

The carbides and nitrides of vanadium and titanium crystallize in the same face centered cubic (fee) system, and because of the closeness of their cell parameters (Table 15.1) form solid solutions. These ceramic materials exhibit interesting mechanical, thermal, chemical and conductive properties.1,2 Their high melting point, hardness and wide range of composition have therefore attracted considerable attention in the last decade. Moreover, their good abrasion resistance and low friction also make these ceramics attractive for protective coating applications.3-5 Chemical vapor deposition (CVD) is a commonly used technique for the production of such materials. In the conventional thermally activated process, a mixture of gases is used.6-9 In the case of TiC, TiN, VC and VN, this mixture is... 

The expansion coefficient of a solid can be estimated with the aid of an approximate thermodynamic equation of state for solids that equates the thermal expansion coefficient P with the quantity yCvp/B, where y is the Griineisen dimensionless ratio. C the specific heat of the solid, p the density of the material, and B the bulk modulus. For face-centered cubic (fee) metals, the average value of the Griineisen constant is 2.3. However, there is a tendency for this constant to increase with atomic number. 

The mixed-metal oxide spinel, MgA fTt, is one of the most important inorganic materials. The structure of spinel can be regarded as a ccp structure of O2-anions with Mg2+ ions orderly occupying /8 of the tetrahedral interstices, and Al3+ ions orderly occupying half of the octahedral interstices the remainder 7/x tetrahedral interstices and half octahedral interstices are unoccupied. The sites of the three kinds of ions in the face-centered cubic unit cell are displayed in Fig. 9.6.29. 

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