(a) Derive planar density expressions for FCC (100) and (111) planes in terms of the atomic radius \(R.\) (b) Compute and compare planar density values for these same two planes for aluminum.

Question: Derive the planar density expressions for FCC (100) and (111) planes in terms of atomic radius R and compute and compare the planar density values for these planes for aluminum. Solution: The planar density expressions for FCC (100) and (111) planes are \(\rho_{100} = \frac{1 \text{ atom}}{(2R)^2}\) and \(\rho_{111} = \frac{\frac{5}{2} \text{ atoms}}{2\sqrt{3}R^2}\), respectively. For aluminum with an atomic radius of 1.43 Å, the planar density values are: \(\rho_{100} = 0.1236 \, \text{atoms/Å}^2\) \(\rho_{111} = 0.2132 \, \text{atoms/Å}^2\) Comparison of planar density values for aluminum planes shows that \(\rho_{111} > \rho_{100}\), which means aluminum's FCC (111) plane is more closely packed than its FCC (100) plane.