What is the packing efficiency of a face-centered cubic unit cell? Please show your work.

Verified step by step guidance

1. Understand that packing efficiency is the fraction of volume in a crystal structure that is occupied by the constituent particles (atoms, ions, or molecules).
2. Recognize that in a face-centered cubic (FCC) unit cell, there are 4 atoms per unit cell. This is calculated as: 8 corner atoms × 1/8 (each corner atom is shared by 8 unit cells) + 6 face atoms × 1/2 (each face atom is shared by 2 unit cells) = 4 atoms.
3. Recall that the volume of a sphere (atom) is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the atom.
4. Calculate the volume of the FCC unit cell. The edge length \( a \) of the FCC unit cell is related to the atomic radius \( r \) by the equation \( a = 2\sqrt{2}r \). Therefore, the volume of the unit cell is \( a^3 = (2\sqrt{2}r)^3 \).
5. Determine the packing efficiency by dividing the total volume of the atoms in the unit cell by the volume of the unit cell, and then multiply by 100 to express it as a percentage. The formula is: \( \text{Packing Efficiency} = \left( \frac{4 \times \frac{4}{3} \pi r^3}{(2\sqrt{2}r)^3} \right) \times 100 \).