Consider the unit cells shown here for three different structures that are commonly observed for metallic elements. (a) Which structure(s) corresponds to the densest packing of atoms?

A unit cell is the smallest repeating unit of a crystal lattice that reflects the overall symmetry and structure of the entire crystal. It defines the arrangement of atoms in a solid and can vary in shape, such as cubic, tetragonal, or hexagonal. Understanding unit cells is crucial for analyzing the packing efficiency and density of different crystalline structures.

Atomic packing efficiency (APE) refers to the fraction of volume in a crystal structure that is occupied by atoms. It is calculated by comparing the volume occupied by the atoms to the total volume of the unit cell. Higher packing efficiency indicates a denser arrangement of atoms, which is important for understanding the physical properties of metals and their stability.

Close-packed structures, such as face-centered cubic (FCC) and hexagonal close-packed (HCP), are arrangements where atoms are packed together as closely as possible. These structures maximize atomic density and are characterized by specific coordination numbers and packing efficiencies. Identifying these structures is essential for determining which metallic arrangements provide the densest packing.