Cubic lattice, packing efficiency, 175

Crystalline solids consist of periodically repeating arrays of atoms, ions or molecules. Many catalytic metals adopt cubic close-packed (also called face-centred cubic) (Co, Ni, Cu, Pd, Ag, Pt) or hexagonal close-packed (Ti, Co, Zn) structures. Others (e.g. Fe, W) adopt the slightly less efficiently packed body-centred cubic structure. The different crystal faces which are possible are conveniently described in terms of their Miller indices. It is customary to describe the geometry of a crystal in terms of its unit cell. This is a parallelepiped of characteristic shape which generates the crystal lattice when many of them are packed together. 

Figure 11.16(b) shows how the cep structure is identical to the face-centered cubic Bravais lattice. In this case, each of the eight corner atoms is shared by eight different unit cells and the six face-centers are each shared by two unit cells. Hence, there are a total of four atoms per unit cell within the cep structure. While the cep contains more atoms per unit cell than the hep, some of its edges are also longer, so that the two closest-packed structures have identical densities and packing efficiencies. Many—but not all—metallic solids will also assume one of the two closest-packed structures. 

In Example I I -2, it was proved that the packing efficiency for the fece-centered cubic Bravais lattice is 74%. What is the packing efficiency of the body-centered cubic Bra-vais lattice Show all work. 

I describe the arrangement of atoms in the common cubic crystal lattices and calculate the packing efficiency for a lattice. 

Treating the individual atoms as spheres of the same radius, we would find that the most efficient packing, filling 74% of the available space, occurs for two patterns, shown in Fig. 13.8 the face-centered cubic (fee) or cubic close-packed (cep) lattice, and the hexagonal close-packed (hep) lattice. The hep lattice can be visualized by arranging one set of atoms in staggered rows on a plane, then adding an identical plane of atoms on top of the first one... 

The face-centered cubic (fee) or cubic close-packed (cep) lattice is a crystal lattice geometry based on a face-centered cubic unit cell, and it is capable of the highest possible packing efficiency for perfect spheres. 

C. The body-centered cubic lattice is the least-efficient packing structure of the metals. What elements in Groups 1 and 2 show this arrangement ... 

Crystalline solids consist of particles tightly packed into a regular array called a crystal lattice. The unit cell is the simplest portion of the crystal that, when repeated, gives the crystal. Many substances crystallize in one of three types of cubic unit cells, which differ in the arrangement of the particles and, therefore, in the number of particles per unit cell and how efficiently they are packed. 

Example 11 -2. Prove that the atoms of both the cubic and hep lattices occupy the same amount of space and that they both pack with 74% efficiency. 

With this result, we can rationalize the expressions for the fraction of space occupied by the spheres and the density as given in Table 7.1. Note that the bcc lattice is a more efficient way to pack spheres (68% compared with 52%) than the simple cubic. 

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