Select the correct alternative from the choices given:
The packing efficiency of simple cubic structure, body centered cubic structure and face centered cubic structure respectively is :

Options

A. `52.4%, 74%, 68%`
B. `74%, 68%, 52.4%`
C. `52.4%, 68%, 74%`
D. `68%, 74%, 52.4%`

Explanation

To determine the packing efficiency of simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC) structures, we can follow these steps:

### Step 1: Understand 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. The formula for packing efficiency (PE) is given by:

\[ PE = \frac{\text{Volume occupied by atoms}}{\text{Volume of unit cell}} \times 100 \]

### Step 2: Calculate Packing Efficiency for Simple Cubic Structure
1. **Volume occupied by atoms**: In a simple cubic structure, there is 1 atom per unit cell. The volume of one atom (considered as a sphere) is:

\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]

2. **Volume of the unit cell**: The side length \( a \) of the unit cell is related to the radius \( r \) of the atom by:

\[ a = 2r \]

Thus, the volume of the unit cell is:

\[ \text{Volume of unit cell} = a^3 = (2r)^3 = 8r^3 \]

3. **Packing Efficiency Calculation**:

\[ PE = \frac{\frac{4}{3} \pi r^3}{8r^3} \times 100 = \frac{\frac{4}{3} \pi}{8} \times 100 \approx 52.4\% \]

### Step 3: Calculate Packing Efficiency for Body-Centered Cubic Structure
1. **Volume occupied by atoms**: In a BCC structure, there are 2 atoms per unit cell. Thus, the total volume occupied by atoms is:

\[ \text{Volume occupied} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]

2. **Volume of the unit cell**: The relationship between \( a \) and \( r \) in BCC is:

\[ a = \sqrt{3} r \]

Thus, the volume of the unit cell is:

\[ \text{Volume of unit cell} = a^3 = (\sqrt{3} r)^3 = 3\sqrt{3} r^3 \]

3. **Packing Efficiency Calculation**:

\[ PE = \frac{\frac{8}{3} \pi r^3}{3\sqrt{3} r^3} \times 100 \approx 68\% \]

### Step 4: Calculate Packing Efficiency for Face-Centered Cubic Structure
1. **Volume occupied by atoms**: In an FCC structure, there are 4 atoms per unit cell. Thus, the total volume occupied by atoms is:

\[ \text{Volume occupied} = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]

2. **Volume of the unit cell**: The relationship between \( a \) and \( r \) in FCC is:

\[ a = 2\sqrt{2} r \]

Thus, the volume of the unit cell is:

\[ \text{Volume of unit cell} = a^3 = (2\sqrt{2} r)^3 = 16\sqrt{2} r^3 \]

3. **Packing Efficiency Calculation**:

\[ PE = \frac{\frac{16}{3} \pi r^3}{16\sqrt{2} r^3} \times 100 \approx 74\% \]

### Final Results
- Packing efficiency of Simple Cubic: **52.4%**
- Packing efficiency of Body-Centered Cubic: **68%**
- Packing efficiency of Face-Centered Cubic: **74%**

### Conclusion
The packing efficiencies for the three structures are:
- Simple Cubic: 52.4%
- Body-Centered Cubic: 68%
- Face-Centered Cubic: 74%

The correct answer is: **52.4%, 68%, 74%**.