FCC Lattice Packing Fraction Calculation

The packing fraction (or packing efficiency) represents the fraction of the volume within a crystal structure unit cell that is occupied by atoms.

Determining Atoms in FCC Unit Cell

• A face-centered cubic (fcc) unit cell contains atoms at the 8 corners and at the center of each of the 6 faces.
• Total atoms per unit cell = (8 corners $\times$ 1/8 atom/corner) + (6 faces $\times$ 1/2 atom/face) = 1 + 3 = 4 atoms.

Relating Atomic Radius to Lattice Constant

• In an fcc structure, atoms touch along the face diagonal.
• Let 'a' be the lattice constant and 'r' be the atomic radius. The length of the face diagonal is $\sqrt{2}a$.
• This length equals four atomic radii ($4r$). Therefore, $\sqrt{2}a = 4r$.
• Solving for 'r', we get $r = \frac{\sqrt{2}a}{4} = \frac{a}{2\sqrt{2}}$.

Calculating Volume Occupied by Atoms

• The volume of a single atom is $\frac{4}{3}\pi r^3$.
• The total volume occupied by atoms in the fcc unit cell is $4 \times (\frac{4}{3}\pi r^3)$.
• Substituting $r = \frac{a}{2\sqrt{2}}$: Total Volume = $4 \times \frac{4}{3}\pi \left(\frac{a}{2\sqrt{2}}\right)^3$ Total Volume = $\frac{16}{3}\pi \left(\frac{a^3}{8 \times 2\sqrt{2}}\right)$ Total Volume = $\frac{16}{3}\pi \frac{a^3}{16\sqrt{2}}$ Total Volume = $\frac{\pi a^3}{3\sqrt{2}}$

Calculating Packing Fraction

• The volume of the unit cell is $a^3$.
• Packing Fraction (PF) = (Total Volume of Atoms) / (Volume of Unit Cell)
• PF = $\frac{\frac{\pi a^3}{3\sqrt{2}}}{a^3}$
• PF = $\frac{\pi}{3\sqrt{2}}$
• Evaluating numerically: PF $\approx \frac{3.14159}{3 \times 1.41421} \approx 0.74048$

The packing fraction for an fcc lattice is approximately 0.74.