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Concept
Atomic packing factor
Summary
In crystallography, atomic packing factor (APF), packing efficiency, or packing fraction is the fraction of volume in a crystal structure that is occupied by constituent particles. It is a dimensionless quantity and always less than unity. In atomic systems, by convention, the APF is determined by assuming that atoms are rigid spheres. The radius of the spheres is taken to be the maximum value such that the atoms do not overlap. For one-component crystals (those that contain only one type of particle), the packing fraction is represented mathematically by
where Nparticle is the number of particles in the unit cell, Vparticle is the volume of each particle, and Vunit cell is the volume occupied by the unit cell. It can be proven mathematically that for one-component structures, the most dense arrangement of atoms has an APF of about 0.74 (see Kepler conjecture), obtained by the close-packed structures. For multiple-component structures (such as with interstitial alloys), the APF can exceed 0.74.
The atomic packing factor of a unit cell is relevant to the study of materials science, where it explains many properties of materials. For example, metals with a high atomic packing factor will have a higher "workability" (malleability or ductility), similar to how a road is smoother when the stones are closer together, allowing metal atoms to slide past one another more easily.
Common sphere packings taken on by atomic systems are listed below with their corresponding packing fraction.
Hexagonal close-packed (HCP): 0.74
Face-centered cubic (FCC): 0.74 (also called cubic close-packed, CCP)
Body-centered cubic (BCC): 0.68
Simple cubic: 0.52
Diamond cubic: 0.34
The majority of metals take on either the HCP, FCC, or BCC structure.
For a simple cubic packing, the number of atoms per unit cell is one. The side of the unit cell is of length 2r, where r is the radius of the atom.
For a face-centered cubic unit cell, the number of atoms is four. A line can be drawn from the top corner of a cube diagonally to the bottom corner on the same side of the cube, which is equal to 4r.
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