Lesson 2, Topic 10
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Atomic Packing Factor

Abdulaziz July 26, 2020
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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.

a is the lattice constant
  • Number of atoms per cell: 4
  • Volume of each atom: 4/3 π R3
  • Unit cell volume: a3 = (22R)3

number of atoms x volume of each atom / unit cell volume

Atomic Packing factor for different structures

StructureRadiusAtom/unit cellAPF
Simple cubica/21π/6 = 52%
BCC(3a)/42(π√3)/8 = 68%
FCC(2a)/44(π√2)/6 = 74%
Diamond(3a)/88(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