Which lattice type corresponds to a = 4R/√3?

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Multiple Choice

Which lattice type corresponds to a = 4R/√3?

Explanation:
In a lattice of hard-sphere atoms, the lattice parameter a is tied to how far apart neighboring atom centers are when the spheres touch. For a body-centered cubic arrangement, the center atom touches the corner atom along the body diagonal of the cube. The centers lie along that diagonal with separation distance equal to sqrt[(a/2)^2+(a/2)^2+(a/2)^2] = (√3/2) a. Since touching means this distance equals 2R, you get (√3/2) a = 2R, giving a = 4R/√3. This geometry is unique to BCC, whereas simple cubic would give a = 2R, FCC would give a = 2√2 R, and hexagonal close-packed has a different relation in the basal plane.

In a lattice of hard-sphere atoms, the lattice parameter a is tied to how far apart neighboring atom centers are when the spheres touch. For a body-centered cubic arrangement, the center atom touches the corner atom along the body diagonal of the cube. The centers lie along that diagonal with separation distance equal to sqrt[(a/2)^2+(a/2)^2+(a/2)^2] = (√3/2) a. Since touching means this distance equals 2R, you get (√3/2) a = 2R, giving a = 4R/√3. This geometry is unique to BCC, whereas simple cubic would give a = 2R, FCC would give a = 2√2 R, and hexagonal close-packed has a different relation in the basal plane.

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