In a face-centered cubic lattice, the lattice parameter a relates to the atomic radius R by which expression?

• A. a = 2R
• B. a = 4R/√3
• C. a = 4R/√2
• D. a = √2 R

Answer: (D)

In an FCC lattice, closest packing occurs along the face diagonal, where neighboring atoms touch. Look at a corner atom and the face-centered atom on the same face. The centers of these two touching atoms are separated by a/√2, since the distance from a corner to the face center on that face is the result of moving a/2 in two directions, giving sqrt((a/2)^2 + (a/2)^2) = a/√2. Because the two touching atoms each have radius R, the center-to-center distance equals 2R. Therefore 2R = a/√2, which leads to a = 2√2 R. This is equivalent to a = 4R/√2 or a = 2√2 R. So the lattice parameter is proportional to the atomic radius by the factor 2√2.