In a multi layered close-packed structure

Options

A. there ae twice as many tetrahedral holes as many close-packed atoms
B. there are as many tetrahedral holes as many closed packed atoms
C. there are twice as many octahedral holes as many close-packed atoms
D. there are as many tetrahedral holes as many octahedral holes

Answer

To solve the question regarding the relationship between tetrahedral and octahedral voids in a multi-layered close-packed structure, we will analyze the structure of a face-centered cubic (FCC) lattice, which is a common example of a close-packed arrangement.

### Step-by-Step Solution:

1. **Identify the Structure**:
- We are dealing with a face-centered cubic (FCC) lattice. In this structure, atoms are located at each of the corners and at the centers of each face of the cube.

2. **Calculate the Number of Atoms in the Unit Cell**:
- In an FCC unit cell, there are:
- 8 corner atoms, each contributing \( \frac{1}{8} \) of an atom (since each corner is shared by 8 unit cells).
- 6 face-centered atoms, each contributing \( \frac{1}{2} \) of an atom (since each face is shared by 2 unit cells).
- Total contribution:
\[ \text{Total atoms} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \text{ atoms} \]

3. **Determine the Number of Tetrahedral Voids**:
- Tetrahedral voids are located along the body diagonals of the cube.
- There are 4 body diagonals in a cube, and each body diagonal contains 2 tetrahedral voids.
- Therefore, the total number of tetrahedral voids is:
\[ \text{Total tetrahedral voids} = 4 \text{ body diagonals} \times 2 = 8 \text{ tetrahedral voids} \]

4. **Determine the Number of Octahedral Voids**:
- Octahedral voids are located at the center of the cube and at the edge centers.
- There are 12 edges in a cube, and each edge contributes \( \frac{1}{4} \) of an octahedral void (since each edge is shared by 4 unit cells).
- The contribution from the edges:
\[ \text{Contribution from edges} = 12 \times \frac{1}{4} = 3 \]
- Adding the body center (which contributes 1):
\[ \text{Total octahedral voids} = 3 + 1 = 4 \text{ octahedral voids} \]

5. **Establish Relationships**:
- From the calculations:
- Number of atoms in the unit cell (n) = 4
- Number of tetrahedral voids = 8 (which is \( 2n \))
- Number of octahedral voids = 4 (which is \( n \))
- Thus, the relationships are:
- There are twice as many tetrahedral voids as there are atoms in the unit cell.
- There are as many octahedral voids as there are atoms in the unit cell.

6. **Evaluate the Options**:
- The options provided in the question can now be evaluated based on our findings:
- Option 1: There are twice as many tetrahedral voids as many closed pack atoms. **(Correct)**
- Option 2: There are as many tetrahedral voids as many closed pack atoms. **(Incorrect)**
- Option 3: There are twice as many octahedral voids as many closed pack atoms. **(Incorrect)**
- Option 4: There are as many tetrahedral voids as octahedral voids. **(Incorrect)**

### Conclusion:
The correct answer is **Option 1**: There are twice as many tetrahedral voids as many closed pack atoms.