For a piece of coursework I am doing, I need to calculate the atomic packing factor of some ionic compounds. I have had no formal teaching in this area, so what I know comes from information I have found on the internet.
I take that the atomic packing factor is given by: $$ APF = \frac{V_\text{atoms}}{V_\text{unit cell}} $$ Taking a unit cell of $\ce{NaCl}$, we get: $$ APF = \frac{2\pi(r_\ce{Cl}^3 + r_\ce{Na}^3)}{3(r_\ce{Cl} + r_\ce{Na})^3} $$ Is this right? I took the side length of the unit cell to be $r_\ce{Cl} + r_\ce{Na}$, and reasoned that 1/8 of each ion is actually "inside" the unit cell, so there is essentially 1/2 of a chloride ion and 1/2 of a sodium ion "within" the cell. $r_\ce{Cl}$ is the ionic radius of chloride, and $r_\ce{Na}$ is that of sodium.
Would this also hold for all alkali metal-halides with a FCC crystal structure? For instance, for a compound with anion $\ce{A}$ and cation $\ce{C}$:
$$ APF = \frac{2\pi(r_\ce{A}^3 + r_\ce{C}^3)}{3(r_\ce{A} + r_\ce{C})^3} $$
Finally, is this a good measure of packing efficiency? As I mentioned, I have little experience in this topic, so I would be grateful if anyone knew of a better measure.