In a bcc structure, the packing fraction is:

Concept:

• The main characteristic of crystalline solids is a regular and repeating pattern of constituent particles.
• If the three-dimensional arrangement of constituent particles in a crystal is represented diagrammatically, in which each particle is depicted as a point, the arrangement is called a crystal lattice.
• Thus, a regular three-dimensional arrangement of points in space is called a crystal lattice.
• The unit cell is the smallest portion of a crystal lattice which, when repeated in different directions, generates the entire lattice.
• In BCC (body-centered cubic) crystal, the arrangement will be as follows 

In the crystal structure, atomic packing factor (APF) or packing efficiency or packing fraction is the volume of atoms in a unit cell divided by the volume of the unit cell. It is a dimensionless quantity and always less than unity

No of atoms in b.c.c unit cell = 2

For BCC \(a = \frac{{4r}}{{\sqrt 3 }}\) where a is the side of the cube and r is the atomic radius.

Now APF will be 

\(APF = \frac{{{N_{atoms}}{V_{atom}}}}{{{V_{crystal}}}} = \frac{{2\left( {\frac{4}{3}} \right)\pi {r^3}}}{{{{\left( {a } \right)}^3}}}= \frac{{2\left( {\frac{4}{3}} \right)\pi {r^3}}}{{{{\left( {\frac{{4r}}{{\sqrt 3 }}} \right)}^3}}}\)

\(APF = \frac {\sqrt {3} \pi}{8}\)

Additional Information