the correct order of the packing effeciency in different types of unit cells is ……….. .

Options

A. fccltBcclt simple cubic
B. fccgt bcc gt simple cubic
C. fcclt bcc? Simple cubic
D. bcclt fccgt simple cubic

Explanation

To determine the correct order of packing efficiency in different types of unit cells, we will calculate the packing efficiency for three types of unit cells: Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC).

### Step 1: Calculate Packing Efficiency for Simple Cubic (SC)

1. **Identify the parameters**:
- Edge length of the unit cell = \( a \)
- Radius of the sphere = \( r \)

2. **Relation between edge length and radius**:
- For SC, spheres touch each other along the edge, so:
\[ a = 2r \quad \Rightarrow \quad r = \frac{a}{2} \]

3. **Number of spheres in the unit cell**:
- There are 8 spheres at the corners, each shared by 8 unit cells:
\[ \text{Number of spheres} = 8 \times \frac{1}{8} = 1 \]

4. **Volume of the sphere**:
\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi \left(\frac{a}{2}\right)^3 = \frac{4}{3} \pi \frac{a^3}{8} = \frac{\pi a^3}{6} \]

5. **Volume of the unit cell**:
\[ V_{\text{cube}} = a^3 \]

6. **Packing efficiency**:
\[ \text{Packing Efficiency} = \frac{V_{\text{occupied}}}{V_{\text{total}}} = \frac{\frac{\pi a^3}{6}}{a^3} = \frac{\pi}{6} \approx 0.524 \quad \text{or} \quad 52.4\% \]

### Step 2: Calculate Packing Efficiency for Body-Centered Cubic (BCC)

1. **Identify the parameters**:
- Edge length of the unit cell = \( a \)
- Radius of the sphere = \( r \)

2. **Number of spheres in the unit cell**:
- 8 corner spheres (each contributes \( \frac{1}{8} \)) and 1 sphere in the center:
\[ \text{Number of spheres} = 8 \times \frac{1}{8} + 1 = 2 \]

3. **Volume of the sphere**:
\[ V_{\text{sphere}} = 4 \pi r^3 \]

4. **Relation between edge length and radius**:
- For BCC, the body diagonal \( \sqrt{3}a \) is equal to 4 times the radius:
\[ \sqrt{3}a = 4r \quad \Rightarrow \quad r = \frac{\sqrt{3}}{4}a \]

5. **Volume of the unit cell**:
\[ V_{\text{cube}} = a^3 \]

6. **Packing efficiency**:
\[ V_{\text{occupied}} = 2 \times \frac{4}{3} \pi r^3 = 2 \times \frac{4}{3} \pi \left(\frac{\sqrt{3}}{4}a\right)^3 = \frac{2 \cdot 4 \cdot 3\sqrt{3}}{3 \cdot 64} \pi a^3 = \frac{3\sqrt{3}}{32} \pi a^3 \]
\[ \text{Packing Efficiency} = \frac{\frac{3\sqrt{3}}{32} \pi a^3}{a^3} \approx 0.6805 \quad \text{or} \quad 68.05\% \]

### Step 3: Calculate Packing Efficiency for Face-Centered Cubic (FCC)

1. **Identify the parameters**:
- Edge length of the unit cell = \( a \)
- Radius of the sphere = \( r \)

2. **Number of spheres in the unit cell**:
- 8 corner spheres and 6 face-centered spheres:
\[ \text{Number of spheres} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4 \]

3. **Relation between edge length and radius**:
- For FCC, the face diagonal \( \sqrt{2}a \) is equal to 4 times the radius:
\[ \sqrt{2}a = 4r \quad \Rightarrow \quad r = \frac{\sqrt{2}}{4}a \]

4. **Volume of the unit cell**:
\[ V_{\text{cube}} = a^3 \]

5. **Packing efficiency**:
\[ V_{\text{occupied}} = 4 \times \frac{4}{3} \pi r^3 = 4 \times \frac{4}{3} \pi \left(\frac{\sqrt{2}}{4}a\right)^3 = \frac{4 \cdot 4 \cdot 2\sqrt{2}}{3 \cdot 64} \pi a^3 = \frac{2\sqrt{2}}{3} \pi a^3 \]
\[ \text{Packing Efficiency} = \frac{\frac{2\sqrt{2}}{3} \pi a^3}{a^3} \approx 0.7406 \quad \text{or} \quad 74.06\% \]

### Final Order of Packing Efficiency

Now we can summarize the packing efficiencies:
- Simple Cubic (SC): 52.4%
- Body-Centered Cubic (BCC): 68.05%
- Face-Centered Cubic (FCC): 74.06%

Thus, the correct order of packing efficiency is:
\[ \text{FCC} > \text{BCC} > \text{SC} \]