What is the general expression for the atomic packing factor (APF) in terms of atomic radius R and lattice parameter a?

• A. (4/3)πR^3/a^3
• B. (4/3)πR^3/(2a)^3
• C. πR^3/a^3
• D. 4πR^3/(3a^3)

Answer: (A)

The general idea tested is how much of a unit cell’s volume is actually filled by atomic spheres. In a simple cubic arrangement there is one whole atom per unit cell because eight corner atoms contribute 1 full atom in total. The volume of that atom is (4/3)πR^3, while the unit cell volume is a^3 since the edge length is a. So the atomic packing factor is the ratio of these volumes: APF = (4/3)πR^3 / a^3. If the atoms touch along the cube edge (R = a/2), this becomes APF = (4/3)π(a/2)^3 / a^3 = π/6 ≈ 0.524. The expression (4/3)πR^3 / a^3 is the correct general form, reflecting the actual occupied volume divided by the cell volume.