Use of the Primitive Unit Cell in Understanding Subtle Features of the Cubic Close-Packed Structure

One of the most important crystal structures adopted by metals is characterized by the abcabc… stacking of close-packed layers. This structure is commonly referred to in textbooks as the cubic close-packed (ccp) or face-centered cubic (fcc) structure, since the entire lattice can be generated by replication of a face-centered cubic unit cell containing exactly four atomic spheres. However, there is an inconsistency between the definition of unit cell generally applied in chemistry textbooks, as the smallest repeating structural unit, and the choice of the face-centered cube. The ccp structure may also be described by a primitive unit cell (containing one atom) that is a rhombohedron, where each face is a rhombus with acute angles of 60 degrees. While the fcc unit cell retains the symmetry of the ccp lattice, there are some clear pedagogical advantages in presenting the primitive rhombohedral unit cell that may help students gain greater insight into the nature of the ccp structure. We have compared the two unit cells and have performed calculations on the rhombohedral unit cell to determine the individual atomic sphere fraction enclosed by each of the vertices, which, as far as we have been able to determine, have not been previously reported.