The face-centered cubic (FCC) unit cell is a three-dimensional arrangement of atoms that features an atom at each of the eight corners of the cube and an atom at the center of each of the six faces. This configuration results in a total of four atoms per unit cell. The contribution of atoms from the corners and faces is crucial for understanding the structure. Each corner atom is shared among eight neighboring unit cells, contributing \(\frac{1}{8}\) of an atom per unit cell. With eight corners, this totals to one atom from the corners. The face-centered atoms are shared between two unit cells, contributing \(\frac{1}{2}\) of an atom each. Since there are six faces, this adds another three atoms, leading to a total of four atoms in the FCC unit cell.

When calculating the edge length of the FCC unit cell, it is important to note that it is not simply \(2r\) (where \(r\) is the atomic radius). Instead, the edge length \(a\) is given by the formula:

\[ a = 2\sqrt{2}r \]

This adjustment accounts for the arrangement of atoms within the unit cell. As the complexity of the structure increases, both the packing efficiency and coordination number also rise. For the FCC unit cell, the packing efficiency is approximately 74%, indicating that 74% of the volume is occupied by atoms. The coordination number, which represents the number of nearest neighbor atoms surrounding a given atom, is 12 in this case. These key values—total atoms, edge length, packing efficiency, and coordination number—are essential when studying face-centered cubic unit cells.