Which expression correctly represents the APF of a body-centered cubic lattice, given two atoms per cell and r = (√3 a)/4?

• A. (4/3)π((√3 a/4)^3)/a^3
• B. (8/3)π((√3 a/4)^3)/a^3
• C. 2(4/3)π((√3 a/4)^3)/a^3
• D. 4(4/3)π((√3 a/4)^3)/a^3

Answer: (C)

APF is the fraction of the unit cell volume that is actually occupied by atoms. For a body-centered cubic lattice, there are two atoms per unit cell, so the total atomic volume is 2 times the volume of one atom, which is (4/3)π r^3. The unit cell volume is a^3, so the APF is [2(4/3)π r^3]/a^3. The given relation r = (√3 a)/4 comes from the atoms touching along the body diagonal, since the body diagonal length is a√3 and equals 4r in BCC. Substituting r into the APF expression gives 2(4/3)π((√3 a/4)^3)/a^3. If you simplify, you obtain APF = π√3/8 ≈ 0.680, which matches the expected packing fraction for a BCC with two atoms per cell.