We now look at cubic unit cells. First, let us do a little math. Consider a cube. It has 6 faces, 12 edges, and 8 vertices. Each edge has a length which we shall call a. Again, consider the three cubic unit cells, now shown.
V = a3.
The next, rather primitive, drawing is of the face of a simple cubic unit cell.
V = a3 = (2r)3 = 8r3.
Let us now look at a cube in general. The following picture is as good as any.
and, in turn, it can be shown that the inner diagonal is just
.
Now, let us look at the face of a face-centered-cubic unit cell. The following plain picture shows all we need to know.
Similarly, with a body-centered cubic lattice with the three touching atoms on the inner diagonal, we see immediately that
This relationship is shown clearly in the next figure.
We can use these various expressions
to relate the density of a metal to its atomic size, to calculate the size
of a unit cell, or even to determine the type of crystal structure a given
metal has. Examples of some of these things are shown with the problems
and some discussion is given in the next section.