Calculate the maximum packing fraction of the unit cell volume that can be filled by hard spheres in the SC, BCC, FCC, Diamond and Hexagonal structures. Which structure most efficiently fills space? SC : It is easy to determine that there is 1 lattice point per unit cell. The maximum radius that a hard sphere can have is , where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch each other. Hence the packing fraction is: B.C.C : It is easy to determine that there is 2 lattice points per unit cell. The maximum radius that a hard sphere can have is , where a is the lattice constant. When the radius is this value, then the spheres on the corners of the unit cell just touch the body-centered sphere. Hence the packing fraction is: F.C.C : It is easy to determine that there is 4 lattice points per unit cell. The maximum radius that a hard sphere can have is , where a is the lattice
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