From Wikipedia,
the free encyclopedia
In
crystallography, atomic
packing factor 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. It is represented
mathematically by
-
where N is the number of atoms in
the crystal and V is the volume.
It can be proven mathematically
that one-component (one type of
atom) close-packed structures,
those that have the most dense
arrangement of atoms, has an APF
of 0.74. In reality, this number
can be higher given specific
intermolecular factors. For
multiple-component structures, the
APF can exceed 0.74.
Worked example
BCC structure
The body-centered cubic crystal
structure contains eight atoms on
each corner of the cube and one
atom in the center. Because the
volume of the corner atoms are
shared between adjacent cells,
each BCC crystal only contains two
whole 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*√2. 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, it becomes
possible to calculate the APF.
-
-
-
-
-
APF of common structures
By identical procedures, the
ideal atomic packing factors of
all crystal structures can be
found. The common ones are
collected here as reference.
- Simple cubic: 0.52
- Body-centered cubic: 0.68
- Hexagonal close-packed: 0.74
- Face-centered cubic: 0.74
References
-
Schaffer, Saxena, Antolovich,
Sanders, and Warner (1999).
The Science and Design of
Engineering Materials,
Second Edition, 81-88, New York:
WCB/McGraw-Hill.
-
Callister, W. (2002).
Materials Science and
Engineering, Sixth Edition,
105-114, San Francisco: John
Wiley and Sons.
