What is the Packing Factor?
Packing factor or Atomic Packing factor (APF) can be defined as the ratio between the volume of the basic atoms of the unit cell (which represents the volume of all atoms in one unit cell ) to the volume of the unit cell itself.
For cubic crystals, it depends on the radius of atoms and the characterization of chemical bondings.
Simple cubic
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.
![{\displaystyle {\begin{aligned}\mathrm {APF} &={\frac {N_{\mathrm {atoms} }V_{\mathrm {atom} }}{V_{\text{unit cell}}}}={\frac {1\cdot {\frac {4}{3}}\pi r^{3}}{\left(2r\right)^{3}}}\\[10pt]&={\frac {\pi }{6}}\approx 0.5236\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a66938286dbb653970c990206dcd99f945ccc268)
Face-centered cubic
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. Using geometry, and the side length, a can be related to r as:
Knowing this and the formula for the volume of a sphere, it becomes possible to calculate the APF as follows:
![{\displaystyle {\begin{aligned}\mathrm {APF} &={\frac {N_{\mathrm {atoms} }V_{\mathrm {atom} }}{V_{\text{unit cell}}}}={\frac {4\cdot {\frac {4}{3}}\pi r^{3}}{\left({2{\sqrt {2}}r}\right)^{3}}}\\[10pt]&={\frac {\pi {\sqrt {2}}}{6}}\approx 0.740\,48048\ .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f98537d25607c30458b1027c44f6f66aa81c834c)
Body-centered cubic
The primitive unit cell for the body-centered cubic crystal structure contains several fractions taken from nine atoms (if the particles in the crystal are atoms): one on each corner of the cube and one atom in the center. Because the volume of each of the eight corner atoms is shared between eight adjacent cells, each BCC cell contains the equivalent volume of two atoms (one central and one on the corner).
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, it becomes possible to calculate the APF as follows:
![{\displaystyle {\begin{aligned}\mathrm {APF} &={\frac {N_{\mathrm {atoms} }V_{\mathrm {atom} }}{V_{\text{unit cell}}}}={\frac {2\cdot {\frac {4}{3}}\pi r^{3}}{\left({\frac {4r}{\sqrt {3}}}\right)^{3}}}\\[10pt]&={\frac {\pi {\sqrt {3}}}{8}}\approx 0.680\,174\,762\,.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b16a2740493531217460ed8aba6c7bc5c3750505)
