The fraction of the unit-cell volume occupied by atoms, treating atoms as hard spheres. About 0.52 for SC, about 0.68 for BCC and about 0.74 for FCC and HCP.
Japanese name: 原子充填率
Atomic packing factor appears in 1 practice question across 1 topic, including Crystal structures and Miller indices.
The atomic packing factor is found as n × (4/3)πr³/V from the number of atoms n in the unit cell, the atomic radius r and the unit cell volume V. There are three steps: (1) count the atoms in the unit cell, allowing for sharing; (2) fix the relation between the lattice parameter a and r from the direction in which atoms touch; (3) calculate the volume ratio. The result is π/6 ≈ 0.52 for SC (a = 2r), √3π/8 ≈ 0.68 for BCC (√3a = 4r) and π/(3√2) ≈ 0.74 for FCC (√2a = 4r). The result does not depend on r and is determined by the structure alone.
In general, structures with larger coordination numbers have higher packing factors, increasing in the order SC (coordination number 6), BCC (8), and FCC and HCP (12). The packing factor, however, only expresses a geometric fraction; density is determined from the atomic mass and unit cell size as ρ = nM/(N_A a³). For the same element, a difference in packing factor shows up as a volume change, as when iron changes from BCC α-iron to FCC γ-iron and the volume per atom decreases by about 1%. Note that it cannot be used to compare the densities of different elements.
The covalently bonded diamond structure has a coordination number of 4 and a packing factor of only √3π/16 ≈ 0.34, yet diamond is one of the hardest substances. The packing factor is a geometric value for packed hard spheres and says nothing about bond strength or directionality. Also, the real size of an atom changes slightly with coordination number, and the apparent radius becomes somewhat smaller as the coordination number decreases. Furthermore, a higher packing factor does not mean that each interstitial site is smaller; relative to the host atom radius, the FCC octahedral site is larger than any interstitial site in BCC.
For ionic crystals, the calculation uses the radii of both cations and anions. For sodium chloride with the NaCl structure (Na⁺ 0.102 nm, Cl⁻ 0.181 nm, a = 0.564 nm), it is about 0.65. The planar packing fraction, which describes how densely atoms are packed on a plane, is found in the same way: π/(2√3) ≈ 0.91 for the FCC {111} plane and about 0.83 for the BCC {110} plane. Planar packing helps explain which planes become slip planes, surface energies, and how etching and crystal growth rates depend on plane orientation.
Answer: B. About 0.74
In FCC, atoms touch along the face diagonal, so √2a = 4r, that is, a = 2√2r. With 4 atoms per unit cell, the packing factor is 4 × (4/3)πr³ / (2√2r)³ = π/(3√2) ≈ 0.74. About 0.68 is the BCC value (atoms touch along the body diagonal, a = 4r/√3, 2 atoms), and about 0.52 is the SC value (a = 2r, 1 atom, π/6). HCP with the ideal c/a also reaches 0.74 and is the other close-packed structure alongside FCC. Note that a high packing factor does not mean that each individual interstitial site is large.
The packing factor is the fraction of the unit cell volume occupied by atoms and is dimensionless. Density is mass per unit volume; even with the same FCC packing factor, aluminum (about 2.70 g/cm³) and copper (about 8.96 g/cm³) differ greatly.
The unit cell holds 8 atoms, which touch along the body diagonal at a distance of a√3/4, giving a packing factor of √3π/16 ≈ 0.34. Silicon and germanium have the same structure, which contains a lot of empty space.
The touching condition fixes a as a constant multiple of r, so both the atomic volume and the unit cell volume scale with r³, and r cancels in the ratio. For the same structure, every element gives the same value.
Body-centered cubic Face-centered cubic Hexagonal close-packed Miller indices