The fraction of the total volume occupied by atoms in a simple cube is

Options

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

Answer

To find the fraction of the total volume occupied by atoms in a simple cubic unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Simple Cubic Unit Cell**:
- In a simple cubic unit cell, there is one atom per unit cell (Z = 1). The atoms are located at the corners of the cube.

2. **Volume of the Cube**:
- The volume (V) of the cube can be calculated using the formula:
\[ V_{\text{cube}} = a^3 \]
where \( a \) is the edge length of the cube.

3. **Volume of the Atom**:
- The volume of a single atom can be approximated as the volume of a sphere:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the atom.

4. **Relationship Between Radius and Edge Length**:
- In a simple cubic structure, the radius \( r \) of the atom is related to the edge length \( a \) of the cube by:
\[ r = \frac{a}{2} \]

5. **Substituting the Radius**:
- Substitute \( r = \frac{a}{2} \) into the volume of the atom:
\[ V_{\text{atom}} = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3 = \frac{4}{3} \pi \frac{a^3}{8} = \frac{4 \pi a^3}{24} = \frac{\pi a^3}{6} \]

6. **Calculating the Fraction of Volume Occupied**:
- The fraction of the total volume occupied by the atom(s) in the unit cell is given by:
\[ \text{Fraction} = \frac{V_{\text{atom}}}{V_{\text{cube}}} = \frac{\frac{\pi a^3}{6}}{a^3} = \frac{\pi}{6} \]

7. **Final Answer**:
- Therefore, the fraction of the total volume occupied by atoms in a simple cubic unit cell is:
\[ \frac{\pi}{6} \]