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

Options

A. `(pi)/(3sqrt(2))`
B. `(pi)/(4sqrt(2))`
C. `pi/4`
D. `pi/6`

Answer

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

### Step-by-Step Solution:

1. **Understand the Structure**:
- In a simple cubic structure, there is one atom per unit cell. This is represented by \( Z = 1 \).

2. **Volume of the Atom**:
- The volume of a single atom (assuming it is spherical) is given by the formula:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the atom.

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

4. **Volume of the Unit Cell**:
- The volume of the cubic unit cell is given by:
\[ V_{\text{cell}} = a^3 \]
Substituting \( a = 2r \) into this equation gives:
\[ V_{\text{cell}} = (2r)^3 = 8r^3 \]

5. **Calculate the Fraction of Volume Occupied**:
- The fraction of the total volume occupied by the atoms in the unit cell is given by:
\[ \text{Fraction} = \frac{\text{Total volume occupied by atoms}}{\text{Total volume of the unit cell}} = \frac{Z \cdot V_{\text{atom}}}{V_{\text{cell}}} \]
Substituting the values we have:
\[ \text{Fraction} = \frac{1 \cdot \frac{4}{3} \pi r^3}{8r^3} \]

6. **Simplify the Expression**:
- Now, we can simplify the fraction:
\[ \text{Fraction} = \frac{\frac{4}{3} \pi r^3}{8r^3} = \frac{4 \pi}{3 \cdot 8} = \frac{4 \pi}{24} = \frac{\pi}{6} \]

7. **Conclusion**:
- The fraction of total volume occupied by the atoms in a simple cubic structure is:
\[ \frac{\pi}{6} \]

### Final Answer:
The fraction of total volume occupied by the atoms in a simple cubic is \( \frac{\pi}{6} \).

---