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

2. **Identify the Effective Number of Atoms (Z)**:
In a simple cubic unit cell, there is 1 atom per unit cell. This is because there is one atom at each of the eight corners of the cube, and each corner atom contributes \( \frac{1}{8} \) of its volume to the unit cell:
\[ Z = 1 \quad (\text{since } 8 \times \frac{1}{8} = 1) \]

3. **Calculate the Volume Occupied by Atoms**:
The volume occupied by the atoms can be calculated using the formula for the volume of a sphere:
\[ \text{Volume of one atom} = \frac{4}{3} \pi r^3 \]
Therefore, the total volume occupied by atoms in the unit cell is:
\[ \text{Total volume occupied} = Z \times \text{Volume of one atom} = 1 \times \frac{4}{3} \pi r^3 = \frac{4}{3} \pi r^3 \]

4. **Determine the Volume of the Unit Cell**:
The volume of the cube (unit cell) is given by:
\[ \text{Volume of cube} = a^3 \]
In a simple cubic structure, the edge length \( a \) is related to the radius \( r \) of the atom by:
\[ a = 2r \]
Thus, the volume of the unit cell becomes:
\[ a^3 = (2r)^3 = 8r^3 \]

5. **Calculate the Packing Efficiency**:
Now, substituting the volumes into the packing efficiency formula:
\[ \text{Packing Efficiency} = \left( \frac{\frac{4}{3} \pi r^3}{8r^3} \right) \times 100 \]
Simplifying this gives:
\[ = \left( \frac{4 \pi}{3 \times 8} \right) \times 100 = \left( \frac{\pi}{6} \right) \times 100 \]
Therefore, the packing efficiency is:
\[ \text{Packing Efficiency} = \frac{\pi}{6} \times 100 = \frac{\pi}{6} \]

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