Which of the following has the smallest packing efficency for atoms of a single type?

Options

A. Body centred cubic
B. simple cubic
C. Face centred cubic
D. none of these

Answer

To determine which of the given cubic systems has the smallest packing efficiency for atoms of a single type, we will calculate the packing efficiency for three types of cubic systems: Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC).

### Step-by-Step Solution:

1. **Understanding Packing Efficiency**:
- Packing efficiency is defined as the ratio of the volume occupied by the atoms in a unit cell to the total volume of the unit cell, expressed as a percentage.
- The formula for packing efficiency (PE) is:
\[ \text{Packing Efficiency} = \left( \frac{\text{Volume occupied by atoms}}{\text{Total volume of the cube}} \right) \times 100 \]

2. **Calculating Packing Efficiency for Simple Cubic (SC)**:
- In a simple cubic unit cell, there are 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom to the unit cell.
- Total number of atoms \( Z \) in SC:
\[ Z = 8 \times \frac{1}{8} = 1 \]
- The edge length \( A \) of the cube is related to the atomic radius \( R \) as:
\[ A = 2R \]
- Volume occupied by the atom:
\[ \text{Volume} = Z \times \frac{4}{3} \pi R^3 = 1 \times \frac{4}{3} \pi R^3 = \frac{4}{3} \pi R^3 \]
- Total volume of the cube:
\[ \text{Total Volume} = A^3 = (2R)^3 = 8R^3 \]
- Packing efficiency for SC:
\[ \text{PE}_{SC} = \left( \frac{\frac{4}{3} \pi R^3}{8R^3} \right) \times 100 = \left( \frac{\frac{4}{3} \pi}{8} \right) \times 100 \approx 52.4\% \]

3. **Calculating Packing Efficiency for Body-Centered Cubic (BCC)**:
- In a BCC unit cell, there are 8 corner atoms and 1 atom at the center.
- Total number of atoms \( Z \) in BCC:
\[ Z = 8 \times \frac{1}{8} + 1 = 2 \]
- The edge length \( A \) of the cube is:
\[ A = \frac{4R}{\sqrt{3}} \]
- Volume occupied by the atoms:
\[ \text{Volume} = Z \times \frac{4}{3} \pi R^3 = 2 \times \frac{4}{3} \pi R^3 = \frac{8}{3} \pi R^3 \]
- Total volume of the cube:
\[ \text{Total Volume} = A^3 = \left(\frac{4R}{\sqrt{3}}\right)^3 = \frac{64R^3}{3\sqrt{3}} \]
- Packing efficiency for BCC:
\[ \text{PE}_{BCC} = \left( \frac{\frac{8}{3} \pi R^3}{\frac{64R^3}{3\sqrt{3}}} \right) \times 100 \approx 68\% \]

4. **Calculating Packing Efficiency for Face-Centered Cubic (FCC)**:
- In an FCC unit cell, there are 8 corner atoms and 6 face-centered atoms.
- Total number of atoms \( Z \) in FCC:
\[ Z = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4 \]
- The edge length \( A \) of the cube is:
\[ A = 2\sqrt{2}R \]
- Volume occupied by the atoms:
\[ \text{Volume} = Z \times \frac{4}{3} \pi R^3 = 4 \times \frac{4}{3} \pi R^3 = \frac{16}{3} \pi R^3 \]
- Total volume of the cube:
\[ \text{Total Volume} = A^3 = (2\sqrt{2}R)^3 = 16\sqrt{2}R^3 \]
- Packing efficiency for FCC:
\[ \text{PE}_{FCC} = \left( \frac{\frac{16}{3} \pi R^3}{16\sqrt{2}R^3} \right) \times 100 \approx 74\% \]

5. **Comparing Packing Efficiencies**:
- Simple Cubic: 52.4%
- Body-Centered Cubic: 68%
- Face-Centered Cubic: 74%
- The smallest packing efficiency is for the **Simple Cubic** structure.

### Final Answer:
The cubic system with the smallest packing efficiency for atoms of a single type is the **Simple Cubic**.