Which lattice type corresponds to a = 4R/√2?

• A. Simple Cubic
• B. Body-Centered Cubic
• C. Face-Centered Cubic
• D. Hexagonal Close-Packed

Answer: (C)

When atoms touch along a direction in a crystal, the distance between lattice points is tied to how the spheres connect along that path. In a face-centered cubic lattice, atoms touch along the face diagonal. The length of a face diagonal is a√2, and along that line you pass through centers of touching spheres: from one corner to the face-centered atom to the opposite corner, totaling 4R in center-to-center distance. Setting a√2 = 4R gives a = 4R/√2 = 2√2 R. That matches the given relation, so the structure is face-centered cubic.

For comparison, in simple cubic the atoms touch along edges, giving a = 2R; in body-centered cubic the touch is along the body diagonal, yielding a = 4R/√3; hexagonal close-packed has a different parameter relationship tied to its hexagonal geometry. The given expression specifically corresponds to the face-centered cubic arrangement.