The packing fraction for a simple cube is

Options

A. 0.74
B. 0.76
C. 0.68
D. 0.52

Answer

To find the packing fraction for a simple cube, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Packing Fraction**:
The packing fraction (PF) is defined as the ratio of the volume occupied by the atoms in the unit cell to the total volume of the unit cell. Mathematically, it is given by:
\[ \text{Packing Fraction (PF)} = \frac{\text{Volume of atoms in unit cell}}{\text{Volume of unit cell}} \]

2. **Volume of Atoms in the Unit Cell**:
In a simple cubic lattice, there is 1 atom per unit cell. The volume of a single atom can be calculated using the formula for the volume of a sphere:
\[ V_{\text{atom}} = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the atom.

3. **Volume of the Unit Cell**:
The volume of the cubic unit cell is given by:
\[ V_{\text{unit cell}} = a^3 \]
where \( a \) is the edge length of the cube.

4. **Relation Between Radius and Edge Length**:
In a simple cubic structure, the atoms are located at the corners of the cube. The distance between the centers of two adjacent corner atoms is equal to the edge length \( a \). Therefore, we have:
\[ a = 2r \quad \Rightarrow \quad r = \frac{a}{2} \]

5. **Substituting Values**:
Now, substituting \( r = \frac{a}{2} \) 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 Packing Fraction**:
Now we can substitute the volumes into the packing fraction formula:
\[ \text{PF} = \frac{V_{\text{atom}}}{V_{\text{unit cell}}} = \frac{\frac{\pi a^3}{6}}{a^3} = \frac{\pi}{6} \]

7. **Numerical Value**:
To find the numerical value, we can use \( \pi \approx 3.14 \):
\[ \text{PF} = \frac{3.14}{6} \approx 0.5233 \]

8. **Final Result**:
Rounding this value gives us approximately 0.52. Therefore, the packing fraction for a simple cube is:
\[ \text{Packing Fraction} \approx 0.52 \]

### Conclusion:
The packing fraction for a simple cube is **0.52**. Hence, the correct option is option 4.