In a face-centered cubic lattice, the edge length a is related to the atomic radius R by which relation?

• A. a = sqrt(2) R
• B. a = 4R
• C. a = R sqrt(8)
• D. a = 2 R sqrt(2)

Answer: (D)

In a face-centered cubic lattice, atoms touch along the face diagonal. On a square face, the centers along that diagonal are a corner atom, a face-centered atom, and the opposite corner atom. Each touching pair is separated by 2R, so the total length along the diagonal is 2R + 2R = 4R. The length of the face diagonal in terms of the edge length a is √2 times a. Setting these equal gives √2 a = 4R, so a = 4R/√2 = 2√2 R, which can also be written as a = R√8 or a = 2R√2. The commonly presented form is a = 2R√2.