Packing in Solids — Core Principles

Core Principles

Packing in solids describes how constituent particles (atoms, ions, molecules) arrange themselves in a crystal lattice to maximize space utilization and achieve stability. Particles are often modeled as hard spheres.

One-dimensional packing is a simple linear arrangement. Two-dimensional packing can be square close-packed (SCP, coordination number 4) or hexagonal close-packed (HCP, coordination number 6). Three-dimensional packing builds upon these layers.

From 2D SCP, we get simple cubic (SC, 52.4% efficiency, CN 6) and body-centered cubic (BCC, 68% efficiency, CN 8). From 2D HCP, we get the most efficient packings: hexagonal close packing (HCP, 74% efficiency, CN 12, ABA stacking) and cubic close packing (CCP or FCC, 74% efficiency, CN 12, ABC stacking).

Empty spaces are called voids, primarily tetrahedral (formed by 4 spheres) and octahedral (formed by 6 spheres). For N atoms in close packing, there are 2N tetrahedral voids and N octahedral voids. Packing efficiency is the percentage of space occupied by particles, a key parameter for material properties.

Important Differences

vs Cubic Close Packing (CCP) / Face-Centered Cubic (FCC)

Aspect	This Topic	Cubic Close Packing (CCP) / Face-Centered Cubic (FCC)
Stacking Sequence	A-B-A-B-...	A-B-C-A-B-C-...
Unit Cell Type	Hexagonal	Cubic (Face-Centered Cubic)
Coordination Number	12	12
Packing Efficiency	74%	74%
Atoms per Unit Cell (Z)	6 (for hexagonal unit cell)	4 (for cubic unit cell)
Examples	Mg, Zn, Ti	Cu, Ag, Au, Al, Ni

While both Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP/FCC) represent the most efficient ways to pack identical spheres in 3D, sharing a packing efficiency of 74% and a coordination number of 12, their fundamental difference lies in their stacking sequence. HCP follows an alternating A-B-A-B pattern, where the third layer directly repeats the first. In contrast, CCP/FCC adopts an A-B-C-A-B-C sequence, meaning the third layer is distinct from the first two, and the fourth layer aligns with the first. This leads to different unit cell geometries: HCP has a hexagonal unit cell, while CCP is described by a face-centered cubic unit cell.