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

Options

A. fcc lt bcc lt simple cubic
B. fcc gt bcc gt simple cubic
C. fcc lt bcc gt simple cubic
D. bcc lt fcc gt simple cubic.

Explanation

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

### Step-by-Step Solution:

1. **Understanding Packing Efficiency**:
- The packing efficiency (PE) is defined as the ratio of the volume occupied by the atoms in the unit cell to the total volume of the unit cell, expressed as a percentage.
- The formula for packing efficiency is:
\[ \text{Packing Efficiency} = \left( \frac{\text{Volume of atoms in unit cell}}{\text{Total volume of unit cell}} \right) \times 100 \]

2. **Packing Efficiency of Simple Cubic (SC)**:
- In a Simple Cubic unit cell, there is 1 atom per unit cell (8 corners × 1/8).
- The relationship between the edge length \(a\) and the radius \(r\) of the atom is:
\[ r = \frac{a}{2} \]
- Volume of the unit cell:
\[ V_{cell} = a^3 = (2r)^3 = 8r^3 \]
- Volume of the atom:
\[ V_{atom} = \frac{4}{3} \pi r^3 \]
- Packing efficiency for SC:
\[ PE_{SC} = \left( \frac{V_{atom}}{V_{cell}} \right) \times 100 = \left( \frac{\frac{4}{3} \pi r^3}{8r^3} \right) \times 100 = \frac{\pi}{6} \times 100 \approx 52.4\% \]

3. **Packing Efficiency of Body-Centered Cubic (BCC)**:
- In a BCC unit cell, there are 2 atoms per unit cell (1 atom at the center and 8 corners × 1/8).
- The relationship between the edge length \(a\) and the radius \(r\) is:
\[ a = \frac{4r}{\sqrt{3}} \]
- Volume of the unit cell:
\[ V_{cell} = a^3 = \left(\frac{4r}{\sqrt{3}}\right)^3 = \frac{64r^3}{3\sqrt{3}} \]
- Volume of the atoms:
\[ V_{atom} = 2 \times \frac{4}{3} \pi r^3 = \frac{8}{3} \pi r^3 \]
- Packing efficiency for BCC:
\[ PE_{BCC} = \left( \frac{V_{atom}}{V_{cell}} \right) \times 100 = \left( \frac{\frac{8}{3} \pi r^3}{\frac{64r^3}{3\sqrt{3}}} \right) \times 100 = \frac{8\pi\sqrt{3}}{64} \times 100 \approx 68\% \]

4. **Packing Efficiency of Face-Centered Cubic (FCC)**:
- In an FCC unit cell, there are 4 atoms per unit cell (8 corners × 1/8 + 6 faces × 1/2).
- The relationship between the edge length \(a\) and the radius \(r\) is:
\[ a = 2\sqrt{2}r \]
- Volume of the unit cell:
\[ V_{cell} = a^3 = (2\sqrt{2}r)^3 = 16\sqrt{2}r^3 \]
- Volume of the atoms:
\[ V_{atom} = 4 \times \frac{4}{3} \pi r^3 = \frac{16}{3} \pi r^3 \]
- Packing efficiency for FCC:
\[ PE_{FCC} = \left( \frac{V_{atom}}{V_{cell}} \right) \times 100 = \left( \frac{\frac{16}{3} \pi r^3}{16\sqrt{2}r^3} \right) \times 100 = \frac{\pi}{\sqrt{2}} \times 100 \approx 74\% \]

5. **Final Order of Packing Efficiency**:
- Based on the calculated packing efficiencies:
- Simple Cubic (SC): 52.4%
- Body-Centered Cubic (BCC): 68%
- Face-Centered Cubic (FCC): 74%
- The correct order of packing efficiency is:
\[ \text{FCC} > \text{BCC} > \text{SC} \]

### Final Answer:
The correct order of packing efficiency in different types of unit cells is:
**FCC > BCC > SC**