Which of the following crystal lattice has the minimum empty space?

Options

A. (a) simple cubic
B. (b) Body centred cubic
C. (c) Face centred cubic
D. (d) Simple tetragonal

Answer

To determine which crystal lattice has the minimum empty space (or void fraction), we will analyze the packing fractions and void fractions of the given crystal lattices: Simple Cubic (SC), Body-Centered Cubic (BCC), Face-Centered Cubic (FCC), and Simple Tetragonal (ST).

### Step-by-Step Solution:

1. **Understanding Crystal Lattices**:
- **Simple Cubic (SC)**: Atoms are located at the corners of the cube.
- **Body-Centered Cubic (BCC)**: Atoms are at the corners and one atom is at the center of the cube.
- **Face-Centered Cubic (FCC)**: Atoms are at the corners and at the centers of each face of the cube.
- **Simple Tetragonal (ST)**: Similar to BCC but with a different height, having a packing fraction similar to BCC.

2. **Calculating Atoms per Unit Cell**:
- **Simple Cubic (SC)**:
- Atoms = 8 corners × (1/8 contribution each) = 1 atom per unit cell.
- **Body-Centered Cubic (BCC)**:
- Atoms = 8 corners × (1/8 contribution each) + 1 body center = 2 atoms per unit cell.
- **Face-Centered Cubic (FCC)**:
- Atoms = 8 corners × (1/8 contribution each) + 6 faces × (1/2 contribution each) = 4 atoms per unit cell.
- **Simple Tetragonal (ST)**:
- Similar to BCC, it has 2 atoms per unit cell.

3. **Calculating Packing Fraction**:
- **Simple Cubic (SC)**: Packing fraction = 52.4%
- **Body-Centered Cubic (BCC)**: Packing fraction = 68%
- **Face-Centered Cubic (FCC)**: Packing fraction = 76%
- **Simple Tetragonal (ST)**: Packing fraction = 69.7%

4. **Calculating Void Fraction**:
- Void fraction = 100% - Packing fraction
- **Simple Cubic (SC)**: Void fraction = 100% - 52.4% = 47.6%
- **Body-Centered Cubic (BCC)**: Void fraction = 100% - 68% = 32%
- **Face-Centered Cubic (FCC)**: Void fraction = 100% - 76% = 24%
- **Simple Tetragonal (ST)**: Void fraction = 100% - 69.7% = 30.3%

5. **Comparing Void Fractions**:
- Simple Cubic (SC): 47.6%
- Body-Centered Cubic (BCC): 32%
- Face-Centered Cubic (FCC): 24%
- Simple Tetragonal (ST): 30.3%

6. **Conclusion**:
- The crystal lattice with the minimum empty space (or void fraction) is **Face-Centered Cubic (FCC)**, which has a void fraction of 24%.

### Final Answer:
**C. Face-Centered Cubic (FCC)** has the minimum empty space.