The efficiency of packing in simple cubic unit cell is

Options

A. `(pi)/(6)`
B. `(sqrt(3)pi)/(8)`
C. `(sqrt(2)pi)/(6)`
D. `(pi)/(16)`

Answer

To find the efficiency of packing in a simple cubic unit cell, we can follow these steps:

### 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.
- Mathematically, it is expressed as:
\[ \text{Packing Efficiency} = \frac{\text{Volume of atoms in unit cell}}{\text{Volume of unit cell}} \times 100\% \]

2. **Identifying Atoms in Simple Cubic Unit Cell**:
- In a simple cubic unit cell, atoms are located only at the corners of the cube.
- There are 8 corners in a cube, and each corner atom contributes \(\frac{1}{8}\) of its volume to the unit cell.
- Therefore, the total number of atoms per unit cell is:
\[ \text{Total atoms} = 8 \times \frac{1}{8} = 1 \text{ atom} \]

3. **Calculating Volume of the Atom**:
- The volume of a single atom (considered as a sphere) is given by:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
- Here, \(r\) is the radius of the atom.

4. **Finding the Edge Length (a) of the Unit Cell**:
- In a simple cubic structure, the edge length \(a\) is related to the radius \(r\) of the atom:
\[ a = 2r \]

5. **Calculating Volume of the Unit Cell**:
- The volume of the unit cell (which is a cube) is given by:
\[ V_{\text{unit cell}} = a^3 = (2r)^3 = 8r^3 \]

6. **Substituting Values into Packing Efficiency Formula**:
- Now, substituting the volumes into the packing efficiency formula:
\[ \text{Packing Efficiency} = \frac{\frac{4}{3} \pi r^3}{8r^3} \]
- Simplifying this expression:
\[ \text{Packing Efficiency} = \frac{4\pi r^3}{3 \times 8r^3} = \frac{4\pi}{24} = \frac{\pi}{6} \]

7. **Final Result**:
- The packing efficiency of a simple cubic unit cell is:
\[ \frac{\pi}{6} \approx 0.5236 \text{ or } 52.36\% \]

### Conclusion:
The efficiency of packing in a simple cubic unit cell is \(\frac{\pi}{6}\).