Which of the following statements are true regarding tatrahedral voids in fcc lattice ?

Options

A. Tetrahedral voids formed by one corner spheres and three face spheres.
B. Two tetrahedral volds can exist on one diagonal body line in fcc cubic
C. Tetrahedral voids are equal to number of spheres involved in body
D. Volume of tetrahedral voids are half of the octahedral voids.

Answer

To determine which statements regarding tetrahedral voids in a face-centered cubic (FCC) lattice are true, we can analyze each statement based on the properties of the FCC structure.

### Step-by-Step Solution:

1. **Understanding the FCC Structure**:
- In an FCC lattice, atoms are located at the corners and the centers of each face of the cube.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell, and each face atom contributes \( \frac{1}{2} \) of an atom.
- There are 8 corner atoms and 6 face atoms in an FCC unit cell.

**Calculation**:
\[ \text{Total atoms in FCC} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \text{ atoms} \]

**Hint**: Remember the contributions of corner and face atoms in an FCC lattice.

2. **Determining the Number of Tetrahedral Voids**:
- In an FCC lattice, there are 8 tetrahedral voids.
- These voids are located along the body diagonals of the cube.
- Each body diagonal contains 2 tetrahedral voids.

**Calculation**:
\[ \text{Total body diagonals} = 4 \text{ (from two corners to one)} \]
\[ \text{Total tetrahedral voids} = 4 \text{ body diagonals} \times 2 \text{ voids per diagonal} = 8 \text{ voids} \]

**Hint**: Visualize the body diagonals in the cube to understand where the tetrahedral voids are located.

3. **Analyzing the Statements**:
- **Statement 1**: A tetrahedral void is formed by one corner atom and three face-centered atoms.
- **True**: This is correct as the tetrahedral void is formed by the arrangement of these atoms.

- **Statement 2**: Two tetrahedral voids can exist on one body diagonal line.
- **True**: As established, there are indeed two tetrahedral voids along each body diagonal.

- **Statement 3**: The number of tetrahedral voids is equal to the number of spheres involved in the body.
- **False**: There are 8 tetrahedral voids, but the number of spheres (atoms) in the FCC unit cell is 4.

- **Statement 4**: The volume of a tetrahedral void is half of the volume of an octahedral void.
- **False**: The volume of a tetrahedral void is not half; it is actually twice that of an octahedral void.

4. **Conclusion**:
- The true statements regarding tetrahedral voids in an FCC lattice are Statement 1 and Statement 2.

### Final Answer:
- **True Statements**:
- Statement 1: True
- Statement 2: True
- **False Statements**:
- Statement 3: False
- Statement 4: False