The percentage of filled spaces in simple cubic lattice is:

Options

A. `36%`
B. `74%`
C. `68%`
D. `52.4%`

Explanation

To find the percentage of filled spaces in a simple cubic lattice, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Simple Cubic Lattice Structure**:
- In a simple cubic lattice, each unit cell has atoms located at each of the eight corners of the cube.

2. **Define Variables**:
- Let \( r \) be the radius of the atom.
- The edge length \( a \) of the cube in a simple cubic lattice is given by the formula:
\[ a = 2r \]

3. **Calculate the Volume of One Atom**:
- The volume \( V \) of one atom (which is modeled as a sphere) is given by the formula:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]

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

5. **Determine the Number of Atoms per Unit Cell**:
- In a simple cubic lattice, there is 1 atom per unit cell (each corner atom contributes \( \frac{1}{8} \) of its volume to the unit cell, and there are 8 corners).

6. **Calculate the Total Volume of Atoms in the Unit Cell**:
- Since there is 1 atom in the unit cell, the total volume of atoms in the unit cell is:
\[ V_{\text{total atoms}} = 1 \times V_{\text{atom}} = \frac{4}{3} \pi r^3 \]

7. **Calculate the Packing Fraction**:
- The packing fraction (the fraction of the volume of the unit cell that is filled by atoms) is given by:
\[ \text{Packing fraction} = \frac{V_{\text{total atoms}}}{V_{\text{cell}}} = \frac{\frac{4}{3} \pi r^3}{8r^3} \]

8. **Simplify the Expression**:
- Simplifying the packing fraction:
\[ \text{Packing fraction} = \frac{\frac{4}{3} \pi}{8} = \frac{\pi}{6} \]

9. **Convert to Percentage**:
- To find the percentage of filled spaces, multiply the packing fraction by 100:
\[ \text{Percentage filled} = \left(\frac{\pi}{6}\right) \times 100 \approx 52.4\% \]

10. **Final Answer**:
- Therefore, the percentage of filled spaces in a simple cubic lattice is approximately **52.4%**.