28.1 The Structure of Solids 1157
thecoordination numberequals 12. There are two ways to accomplish this closest
packing, as depicted in Figure 28.3. If one layer of spheres is laid down in a plane,
another layer can be placed on it, as shown in Figure 28.3a (the lower layer is drawn
in broken curves). There are now two choices for laying down a third layer: either
in the locations markedh(directly over the spheres of the first layer), or in the
locations markedc. If the third layer is laid down in thehlocations, a lattice with a
hexagonal unit cell results, as shown in Figure 28.3b. If the third layer is laid down
in theclocations, a lattice with a face-centered cubic (fcc) unit cell results, as shown
in Figure 28.3c. In the fcc lattice, the layers of lattice points are not parallel to the
unit cell faces, but pass through alternate corners of the unit cell.
Both the hexagonal and the fcc close-packed lattices give the samepacking fraction,
the fraction of the unit cell volume occupied by the spheres, assuming that the spheres
touch each other. (It is also called the “filling fraction” or the “packing efficiency.”)
The nearest-neighbor distance and the number of nearest neighbors are also identical.
However, the spheres beyond the shell of nearest neighbors are not at the same
distances in both lattices, and a given substance at equilibrium will crystallize in only
one of these lattices. Nickel crystallizes in the face-centered cubic lattice, whereasccccchhhhhhhhhhcc(a)(b)(c)Figure 28.3 Closest Packing of Spheres.(a) Layers of spheres. (b) The hexagonal
close-packed unit cell. This figure shows that the third layer has each sphere directly over
a sphere of the first layer. (c) The face-centered cubic close-packed unit cell. This unit cell
is oriented so that the layers of atoms in Figure 28.3a are parallel to a plane that passes
through the lattice point at the top rear of the cell, the lower left of the cell, and the lower
right of the cell.