Question

Calculate the packing capacity (efficiency) of a simple cubic lattice.

Hint:

Remember: Packing efficiency is lowest for simple cubic (52.4%), higher for bcc (68%), and highest for fcc (74%).

Answer 1

Step 1: Atoms per unit cell.
In a simple cubic lattice, each corner atom is shared by 8 unit cells. Thus, number of atoms per unit cell = \(\frac{1}{8} \times 8 = 1\).
Step 2: Relation between atomic radius and edge length.
For simple cubic: \(a = 2r\).
Step 3: Volume of atom and unit cell.
- Volume of one atom = \(\frac{4}{3}\pi r^3\).
- Volume of unit cell = \(a^3 = (2r)^3 = 8r^3\).
Step 4: Packing efficiency.
\[ \text{Packing efficiency} = \frac{\text{Volume of atoms in cell}}{\text{Volume of unit cell}} \times 100 = \frac{\frac{4}{3}\pi r^3}{8r^3} \times 100 \approx 52.4% \] Step 5: Conclusion.
Thus, the packing efficiency of a simple cubic lattice is
52.4%.