Packing fraction of an identical solid sphere is `74%` in :

Options

A. simple cubic structure
B. fcc structure
C. hcp structure
D. bcc structure

Answer

To determine the packing fraction of an identical solid sphere, we need to analyze different types of crystal structures and their packing efficiencies. The packing fraction is defined as the ratio of the volume occupied by the spheres to the total volume of the unit cell.

### Step-by-Step Solution:

1. **Understanding Packing Fraction**:
- The packing fraction (PF) is calculated using the formula:
\[ \text{Packing Fraction} = \frac{Z \cdot V_{\text{sphere}}}{V_{\text{unit cell}}} \]
- Where \( Z \) is the effective number of atoms per unit cell, \( V_{\text{sphere}} \) is the volume of one sphere, and \( V_{\text{unit cell}} \) is the volume of the unit cell.

2. **Volume of a Sphere**:
- The volume of one sphere (solid sphere) is given by:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \]

3. **Simple Cubic Structure**:
- In a simple cubic structure, there is 1 atom per unit cell (Z = 1).
- The edge length \( a \) is related to the radius \( r \) of the sphere as \( a = 2r \).
- The volume of the unit cell is:
\[ V_{\text{unit cell}} = a^3 = (2r)^3 = 8r^3 \]
- Calculating the packing fraction:
\[ \text{PF} = \frac{1 \cdot \frac{4}{3} \pi r^3}{8r^3} = \frac{4\pi}{24} = \frac{\pi}{6} \approx 0.524 \text{ or } 52.4\% \]

4. **Face-Centered Cubic (FCC) Structure**:
- In FCC, there are 4 atoms per unit cell (Z = 4).
- The relationship between edge length and radius is \( a = 2\sqrt{2}r \).
- The volume of the unit cell is:
\[ V_{\text{unit cell}} = a^3 = (2\sqrt{2}r)^3 = 16\sqrt{2}r^3 \]
- Calculating the packing fraction:
\[ \text{PF} = \frac{4 \cdot \frac{4}{3} \pi r^3}{16\sqrt{2}r^3} = \frac{16\pi}{48\sqrt{2}} \approx 0.74 \text{ or } 74\% \]

5. **Hexagonal Close Packing (HCP)**:
- Similar to FCC, HCP also has 6 effective atoms per unit cell (Z = 6).
- The packing fraction for HCP is also calculated to be 74%.

6. **Body-Centered Cubic (BCC) Structure**:
- In BCC, there are 2 atoms per unit cell (Z = 2).
- The relationship between edge length and radius is \( a = \frac{4}{\sqrt{3}}r \).
- The volume of the unit cell is:
\[ V_{\text{unit cell}} = a^3 = \left(\frac{4}{\sqrt{3}}r\right)^3 = \frac{64}{3\sqrt{3}}r^3 \]
- Calculating the packing fraction:
\[ \text{PF} = \frac{2 \cdot \frac{4}{3} \pi r^3}{\frac{64}{3\sqrt{3}}r^3} = \frac{8\pi\sqrt{3}}{64} \approx 0.68 \text{ or } 68\% \]

### Conclusion:
The packing fraction of an identical solid sphere is 74% in both the Face-Centered Cubic (FCC) structure and the Hexagonal Close Packing (HCP) structure.