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Maximum fraction of the volume in a unit cell occupied by the atoms.
- Assume that the atoms are closely packed and that they can be treated as hard spheres.
- This fraction is called atomic packing factor (APF) or packing density.
Eg. Calculate APF for the fcc (cubic closed packed) structure.
- Number of atoms per cell: 4
- Volume of each atom: 4/3 π R3
- Unit cell volume: a3 = (2√2R)3
number of atoms x volume of each atom / unit cell volume
Atomic Packing factor for different structures
| Structure | Radius | Atom/unit cell | APF |
| Simple cubic | a/2 | 1 | π/6 = 52% |
| BCC | (√3a)/4 | 2 | (π√3)/8 = 68% |
| FCC | (√2a)/4 | 4 | (π√2)/6 = 74% |
| Diamond | (√3a)/8 | 8 | (π√3)/16 = 34% |
As expected, fcc has the highest apf since it is essentially a close packed structure which has the most efficient packing
Crystal structure vs material properties
- Previous egs:
- carbon (CNT, graphite, diamond, C60)
- Si (amorphous, single crystal, polycrystalline)
- Fe (α, β, γ, δ, ε):
- ferrite (alpha iron) -forms below 770 °C (Curie point TC); becomes magnetic; BCC
- beta forms below 912°C ; BCC crystal structure
- gamma -forms below 1394 °C; FCC
- delta –forms from cooling down molten iron below 1538°C; BCC crystal structure
- epsilon –forms at high pressures
- Others: boron, Ge, tin all exist in different structures