Packing Efficiency

Packing efficiency is a measure of how efficiently the constituent particles (atoms, ions, or molecules) fill the available space in a crystal lattice. It is defined as the fraction of the total unit cell volume that is actually occupied by atoms, expressed as a percentage. The remaining space, called void space or interstitial space, is left unoccupied. Understanding packing efficiency helps chemists predict physical properties such as density, compressibility, and the sizes of holes available for interstitial atoms in alloys and ionic compounds. Crystalline solids adopt repeating three-dimensional arrangements called unit cells. The three most common unit cell types for metallic elements are Simple Cubic (SC), Body-Centred Cubic (BCC), and Face-Centred Cubic (FCC). Hexagonal Close-Packed (HCP) is a related arrangement that achieves the same packing efficiency as FCC. Each arrangement differs in how many atoms are effectively contained in one unit cell and in the geometric relationship between atomic radius and the cell edge length, which together determine packing efficiency. The calculation of packing efficiency combines basic geometry with the concept of the effective number of atoms per unit cell. Atoms at corners are shared among eight unit cells, atoms on faces are shared among two, and atoms entirely inside a cell belong only to that cell. By comparing the total volume of the spherical atoms with the volume of the cubic (or hexagonal) cell, students can derive the iconic values: ~52% for SC, ~68% for BCC, and ~74% for FCC and HCP — the maximum possible packing for identical spheres, a result proven mathematically by Kepler's conjecture (now a theorem).

Key takeaways
  • Effective Number of Atoms per Unit Cell
  • Relating Atomic Radius (r) to Edge Length (a)
  • Calculating Packing Efficiency
  • Void Percentage and Its Significance
  • HCP Packing Efficiency

Effective Number of Atoms per Unit Cell

Count atoms by their positional contribution. SC: 8 corner atoms × (1/8) = 1 atom. BCC: 8 corners × (1/8) + 1 body-centre × 1 = 2 atoms. FCC: 8 corners × (1/8) + 6 face-centres × (1/2) = 4 atoms. This 'effective atom count' (Z) is the first variable needed.

Relating Atomic Radius (r) to Edge Length (a)

Atoms touch along specific directions that fix the ratio r/a. SC: atoms touch along the edge, so 2r = a, giving r = a/2. BCC: atoms touch along the body diagonal (length a√3), so 4r = a√3, giving r = (√3/4)a. FCC: atoms touch along the face diagonal (length a√2), so 4r = a√2, giving r = (√2/4)a = a/(2√2). These geometric relationships are essential for the volume calculation.

Calculating Packing Efficiency

Packing Efficiency (PE) = (Volume of atoms in unit cell) / (Volume of unit cell) × 100%. Volume of one atom = (4/3)πr³. Volume of unit cell (cubic) = a³. Substituting the r-to-a relationship for each cell type yields: SC: PE = Z·(4/3)π(a/2)³ / a³ = (4/3)π(1/8) = π/6 ≈ 52.36%. BCC: PE = 2·(4/3)π((√3/4)a)³ / a³ = (8π√3)/16 = π√3/8 ≈ 68.02%. FCC: PE = 4·(4/3)π(a/(2√2))³ / a³ = 4·(4/3)π·(1/(16√2)) = π/(3√2) ≈ 74.05%.

Void Percentage and Its Significance

Void percentage = 100% − Packing Efficiency. SC voids ≈ 47.64%, BCC voids ≈ 31.98%, FCC/HCP voids ≈ 25.95%. The void space in BCC and FCC structures accommodates smaller interstitial atoms (e.g., carbon in iron to form steel). FCC has both octahedral voids (radius ratio limit 0.414) and tetrahedral voids (limit 0.225), which is critical in understanding ionic crystal structures and solid solutions.

HCP Packing Efficiency

HCP has a hexagonal unit cell with c/a ≈ 1.633 for ideal close packing. Each HCP unit cell contains an effective Z = 6 atoms (12 corner × 1/6 + 2 face × 1/2 + 3 interior × 1). Despite the different geometry, the packing efficiency works out identically to FCC at 74.05% because both are close-packed arrangements — the only difference is the stacking sequence (ABCABC for FCC, ABABAB for HCP). Many metals (e.g., Mg, Zn) adopt HCP; others (e.g., Cu, Al, Ni) adopt FCC.

The Key Equation

PE=Z⋅43πr3a3×100%\text{PE} = \frac{Z \cdot \dfrac{4}{3}\pi r^3}{a^3} \times 100\%
  • PE — Packing Efficiency (%): Fraction of unit cell volume occupied by atoms, expressed as a percentage.
  • Z — Effective number of atoms (dimensionless): Total atoms attributed to one unit cell after accounting for sharing (SC=1, BCC=2, FCC=4).
  • r — Atomic radius (m or pm): Radius of one atom, modelled as a hard sphere.
  • a — Edge length of unit cell (m or pm): Length of one side of the cubic unit cell; related to r by the contact condition.

Worked Example

Problem. Iron (Fe) crystallises in a BCC structure at room temperature with an atomic radius of 126 pm. (a) Calculate the edge length of the unit cell. (b) Calculate the packing efficiency and void percentage. (c) Calculate the density of iron given its molar mass is 55.85 g/mol.

a=4r3≈291 pm,PE=π38≈68.0%,ρ=ZMNaa3≈7.9 g cm−3a = \frac{4r}{\sqrt{3}} \approx 291\text{ pm},\quad PE = \frac{\pi\sqrt{3}}{8} \approx 68.0\%,\quad \rho = \frac{ZM}{N_a a^3} \approx 7.9\text{ g cm}^{-3}

Answer. Edge length ≈ 291 pm; Packing efficiency ≈ 68.0%; Void space ≈ 32.0%; Calculated density ≈ 7.53–7.90 g/cm³ depending on exact radius used.

Figure
Diagram
Packing Efficiency Comparison: SC, BCC, FCCSimple Cubic (SC)Body-Centred Cubic (BCC)Face-Centred Cubic (FCC)Z = 1 atomr = a/2Z = 2 atomsr = (√3/4)aZ = 4 atomsr = a/(2√2)Packing Efficiency Bar Chart52%SC68%BCC74%FCC/HCP0%80%Void %:SC: 47.6%BCC: 32.0%FCC: 26.0%Void% = 100% − PE%
Origins · 1 minAuguste Bravais and how this was discovered
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Auguste Bravais
1811–1863 · French

Bravais systematically proved in 1850 that there are precisely 14 distinct three-dimensional lattice types (now called Bravais lattices) that can fill space by translation. His mathematical framework provided the foundation for classifying all crystal structures — including SC, BCC, and FCC — into symmetry categories, making rigorous calculation of properties like packing efficiency possible. Bravais was also a distinguished botanist and studied the mathematical arrangement of leaves on plant stems (phyllotaxis) — applying the same geometric thinking he used for crystals to the spiral patterns of nature.

  • Unit Cells and Crystal Lattices — Packing efficiency calculations begin with identifying the unit cell type and correctly counting the effective number of atoms (Z) using lattice geometry.
  • Interstitial Voids in Crystals — Void percentage is the complement of packing efficiency; octahedral and tetrahedral voids in FCC/HCP structures determine the solubility of interstitial atoms in alloys and ionic solids.
  • Density Calculations for Crystal Structures — The formula ρ = ZM/(Nₐa³) uses the same Z and a values derived during packing efficiency calculations to connect microscopic structure to macroscopic density.
  • Radius Ratio Rules in Ionic Solids — The size of interstitial holes (derived from packing geometry) determines which cation can fit in which void, governing the structure of ionic compounds like NaCl and ZnS.
  • Coordination Number in Crystal Structures — Coordination number (SC=6, BCC=8, FCC=12) reflects how tightly atoms are packed; higher coordination number directly correlates with higher packing efficiency.