Simple Cubic Arrangement

Stacking the AAA... two-dimensional layer directly on top of itself, layer after layer -- so that every sphere in one layer sits exactly on top of the sphere immediately below it, with all layers perfectly aligned both horizontally and vertically -- builds the three-dimensional simple cubic structure. In this arrangement, every sphere touches 6 neighbours (4 within its own layer, one directly above, one directly below), so the coordination number of the simple cubic arrangement is 6.

Packing efficiency. Because spheres cannot fill space perfectly, some free volume always remains between them; the fraction of the total volume that IS actually occupied by the spheres is called the packing efficiency (or packing fraction):

For a cube of edge length a, the cube's volume is a^3. Since the spheres (radius r) touch along the cube's edge, a = 2r, so r = a/2. The volume of one sphere is therefore:

Since a simple cubic unit cell contains exactly 1 sphere (Section 6.5.1), the total volume occupied by spheres in the cell is simply pi*a^3/6. Dividing by the cell's volume a^3 and multiplying by 100:

$\text{Packing efficiency} = \dfrac{\pi a^3/6}{a^3}\times 100 = \dfrac{\pi}{6}\times 100 \approx 52.38\%$ …