In crystallography, atomic packing factor (APF) or packing fraction is the fraction of volume in a crystal structure that is occupied by atoms. It is dimensionless and always less than unity. For practical purposes, the APF of a crystal structure is determined by assuming that atoms are rigid spheres. For onecomponent crystals (those that contain only one type of atom), the APF is represented mathematically by
where N_{atoms} is the number of atoms in the unit cell, V_{atom} is the volume of an atom, and V_{unit cell} is the volume occupied by the unit cell. It can be proven mathematically that for onecomponent structures, the most dense arrangement of atoms has an APF of about 0.74. In reality, this number can be higher due to specific intermolecular factors. For multiplecomponent structures, the APF can exceed 0.74.
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The primitive unit cell for the bodycentered cubic (BCC) crystal structure contains nine atoms: one on each corner of the cube and one atom in the center. Because the volume of each corner atom is shared between adjacent cells, each BCC cell contains two atoms.
Each corner atom touches the center atom. A line that is drawn from one corner of the cube through the center and to the other corner passes through 4r, where r is the radius of an atom. By geometry, the length of the diagonal is a√3. Therefore, the length of each side of the BCC structure can be related to the radius of the atom by
Knowing this and the formula for the volume of a sphere((4 / 3)pi r^{3}), it becomes possible to calculate the APF as follows:
For the hexagonal closepacked (HCP) structure the derivation is similar. The side length of the hexagon will be denoted as a while the height of the hexagon will be denoted as c. Then:
It is then possible to calculate the APF as follows:
By similar procedures, the ideal atomic packing factors of all crystal structures can be found. The common ones are collected here as reference, rounded to the nearest hundredth.
