Packing fraction in a body - centred cubic cell of crystals is 

Options

A. `(sqrt3)/(8)pi`
B. `(pi)/(6)`
C. `(sqrt2)/(6)pi`
D. `(1)/(2sqrt2)pi`

Answer

To calculate the packing fraction in a body-centered cubic (BCC) cell, we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Body-Centered Cubic Structure**:
- In a BCC unit cell, there are atoms located at the eight corners of the cube and one atom in the center of the cube.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell because it is shared among eight adjacent unit cells.
- Therefore, the total number of atoms \( n \) in a BCC unit cell is:
\[ n = 8 \times \frac{1}{8} + 1 = 2 \]

2. **Finding the Radius of the Atoms**:
- In a BCC structure, the relationship between the edge length \( a \) of the cube and the radius \( r \) of the atoms is given by:
\[ r = \frac{\sqrt{3}}{4} a \]

3. **Calculating the Volume of the Atoms in the Unit Cell**:
- The volume \( V \) of the atoms in the unit cell can be calculated using the formula for the volume of a sphere:
\[ V = n \times \frac{4}{3} \pi r^3 \]
- Substituting \( n = 2 \) and \( r = \frac{\sqrt{3}}{4} a \):
\[ V = 2 \times \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4} a\right)^3 \]
- Simplifying this:
\[ V = 2 \times \frac{4}{3} \pi \left(\frac{3\sqrt{3}}{64} a^3\right) = \frac{8\pi \cdot 3\sqrt{3}}{192} a^3 = \frac{8\pi \sqrt{3}}{64} a^3 = \frac{\pi \sqrt{3}}{8} a^3 \]

4. **Calculating the Volume of the Unit Cell**:
- The volume of the cubic unit cell is given by:
\[ V_{cell} = a^3 \]

5. **Calculating the Packing Fraction**:
- The packing fraction \( PF \) is defined as the ratio of the volume occupied by the atoms to the volume of the unit cell:
\[ PF = \frac{V_{atoms}}{V_{cell}} = \frac{\frac{\pi \sqrt{3}}{8} a^3}{a^3} \]
- Simplifying this gives:
\[ PF = \frac{\pi \sqrt{3}}{8} \]

6. **Calculating the Percentage Packing Fraction**:
- To express the packing fraction as a percentage, we multiply by 100:
\[ \text{Percentage Packing Fraction} = PF \times 100 = \frac{\pi \sqrt{3}}{8} \times 100 \]

### Final Result:
The packing fraction in a body-centered cubic cell of crystals is:
\[ \frac{\pi \sqrt{3}}{8} \approx 0.68 \quad \text{(or 68% when expressed as a percentage)} \]