The ratio of packing fraction in fcc, bcc, and cubic structure is, respectively,

Options

A. 52%, 68%, 74%
B. 74%, 68%, 52%
C. 68%, 52%, 74%
D. 52%, 74%, 68%

Answer

To solve the question regarding the ratio of packing fraction in face-centered cubic (FCC), body-centered cubic (BCC), and simple cubic (PCC or SC) structures, we will follow these steps:

### Step 1: Understanding Packing Fraction
The packing fraction (or packing efficiency) is defined as the ratio of the volume occupied by the atoms in a unit cell to the total volume of the unit cell. It can be expressed mathematically as:
\[ \text{Packing Fraction} = \frac{\text{Volume occupied by atoms}}{\text{Total volume of unit cell}} \times 100 \]

### Step 2: Calculate Packing Fraction for FCC
1. **Determine the number of atoms (Z)**: In FCC, \( Z = 4 \) (1 atom at each of the 8 corners and 1 atom at each of the 6 faces).
2. **Relation between radius (r) and edge length (a)**: For FCC, \( r = \frac{\sqrt{2}}{4} a \).
3. **Volume occupied by atoms**:
\[ \text{Volume} = Z \times \frac{4}{3} \pi r^3 = 4 \times \frac{4}{3} \pi \left(\frac{\sqrt{2}}{4} a\right)^3 \]
4. **Total volume of the unit cell**:
\[ \text{Total Volume} = a^3 \]
5. **Calculate the packing fraction**:
\[ \text{Packing Fraction}_{\text{FCC}} = \frac{4 \times \frac{4}{3} \pi \left(\frac{\sqrt{2}}{4} a\right)^3}{a^3} \times 100 \]
After simplification, this yields approximately \( 74\% \).

### Step 3: Calculate Packing Fraction for BCC
1. **Determine the number of atoms (Z)**: In BCC, \( Z = 2 \) (1 atom at each corner and 1 atom in the center).
2. **Relation between radius (r) and edge length (a)**: For BCC, \( r = \frac{\sqrt{3}}{4} a \).
3. **Volume occupied by atoms**:
\[ \text{Volume} = Z \times \frac{4}{3} \pi r^3 = 2 \times \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4} a\right)^3 \]
4. **Total volume of the unit cell**:
\[ \text{Total Volume} = a^3 \]
5. **Calculate the packing fraction**:
\[ \text{Packing Fraction}_{\text{BCC}} = \frac{2 \times \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4} a\right)^3}{a^3} \times 100 \]
After simplification, this yields approximately \( 68\% \).

### Step 4: Calculate Packing Fraction for Simple Cubic (PCC)
1. **Determine the number of atoms (Z)**: In simple cubic, \( Z = 1 \) (1 atom at each corner).
2. **Relation between radius (r) and edge length (a)**: For simple cubic, \( r = \frac{1}{2} a \).
3. **Volume occupied by atoms**:
\[ \text{Volume} = Z \times \frac{4}{3} \pi r^3 = 1 \times \frac{4}{3} \pi \left(\frac{1}{2} a\right)^3 \]
4. **Total volume of the unit cell**:
\[ \text{Total Volume} = a^3 \]
5. **Calculate the packing fraction**:
\[ \text{Packing Fraction}_{\text{PCC}} = \frac{1 \times \frac{4}{3} \pi \left(\frac{1}{2} a\right)^3}{a^3} \times 100 \]
After simplification, this yields approximately \( 52\% \).

### Conclusion
The packing fractions for FCC, BCC, and PCC are approximately:
- FCC: 74%
- BCC: 68%
- PCC: 52%

Thus, the ratio of packing fractions in FCC, BCC, and simple cubic structures is:
**74% : 68% : 52%**

### Final Answer
The correct option is Option 2: 74%, 68%, and 52%.