Closest Packed Structures

Consider an assembly of spherical atoms, placed in a layer with atoms touching as shown by the shaded circles in Figure 1.1 (layer A). A second layer of atoms can be laid down above the first layer as shown by the white circles (layer B). Placing the third and fourth layers over A and B gives the pattern ABABAB. . . . This arrangement is called hexagonal closest packing and the unit cell is shown in Figure 1.2.

Figure 1.1. Closest packing of spheres.

Figure 1.2. Hexagonal unit cell.

The hexagonal unit cell is not cubic; the top and bottom are rhombuses and the vertical dimension is c = √8/3 a. The top and bottom atoms are in layer A, the center atom in layer B. 1/6 of each atom at 120° corners is inside the unit cell, 1/12 of each atom at 60° corners is inside the unit cell; the atom in the center of course is entirely in the unit cell. Adding these up, we have 2 atoms/unit cell.

Returning to Figure 1.1, we could also place the third layer atoms over neither of the first two layers, so that one atom is a position C. More layers can be added, following the ABCABC . . . pattern. This arrangement, called cubic closest packing, has an fcc unit cell (the closest packed layers can be seen as planes of atoms perpendicular to the diagonal of the unit cell in Figure 9-2).

The fcc unit cell can be thought of as having holes in which other atoms or ions can be placed. For example, the Na⁺ ions occupy octahedral holes in the fcc Cl^- lattice (connecting the six Cl^- ions surrounding a Na⁺ by lines defines an octahedron). The fcc lattice also has tetrahedral holes; if we divide the unit cell into 8 smaller cubes, the centers of these little cubes are surrounded by 4 lattice points which define a tetrahedron. The diamond allotropic form of carbon has a fcc structure with C atoms in 4 of these tetrahedral sites; each C atom is surrounded by (covalently bound to) 4 other C's.