The definition is easy; it's merely the ratio of the volume of the atoms in the unit cell to the volume of the cell:
To find the PF for a particular crystal, you need to know its structure. Let's do aluminum, for which the structure (FCC) and the lattice parameter (4.04958 Angstroms) are given on the inside front cover of your textbook.
Since the structure is cubic, the lattice parameter is the length of one side of the cube (the dimension ao in Figure 3-7 on page 45). Therefore the cell volume is the cube of this number. As you'll soon see, we don't actually have to have the value of the lattice parameter. From the figure, you can see that the atoms touch along a face diagonal. Therefore the length of the face diagonal is 4r, where r is the radius of an atom. Note that the unit cell only extends to the centers of the corner atoms! Using the Pythagorean Law,
The atomic volume is the volume of one atom times the number of atoms in the unit cell. For the FCC structure, there are eight atoms on the corners and one atom in each of the 6 faces. However, this does not mean that the unit cell contains 14 atoms! Each corner atom is shared by eight nearby cells, so each corner atom contributes only 1/8 atom to the cell, for a total of 1 atom. You can see this easier by studying Figure 3-8 on page 46. Each face-centered atom is shared by one other nearby cell, so each face-centered atom contributes only ½ atom to the cell. Multiplying this by 6, the contribution of the face-centered atoms is 3 atoms. Thus, the FCC structure has 4 atoms per unit cell. Each atom has a volume given by the formula for the volume of a sphere, so...
Now we're ready to plug into the formula for the packing fraction (PF):
As you can see, this result does not depend on the material (aluminum); it is a function of the cell structure alone.
You should make sure that you can do the
same calculation for the simple cubic and BCC structures, for which the
packing factors are given in Table 3.2 on page 48.