Final answer:
The simple cubic lattice has a packing efficiency of about 52%, the body-centered cubic lattice about 68%, and the face-centered cubic lattice about 74%. The diamond lattice, which is similar to the FCC, also has high packing efficiency with a coordination number of 12.
Step-by-step explanation:
The maximum fractions of unit cell volume that can be filled by hard spheres in different lattice structures are determined by the packing efficiency of each structure. In a simple cubic (SC) lattice, the spheres fill about 52% of the volume. A body-centered cubic (BCC) lattice has a slightly higher packing efficiency, with spheres filling about 68% of the total volume. The most efficient of the cubic lattices is the face-centered cubic (FCC) structure, also known as cubic close packing, where spheres fill approximately 74% of the volume. The diamond lattice, which is a variant of the FCC structure with a basis, has a different packing efficiency but is also highly efficient due to its structure. The FCC and diamond lattices share the same coordination number of 12, indicating that each atom is touching 12 other atoms.