The number of octahedral sites in a cubical close pack array of N spheres is

Options

A. N/2
B. 2 N
C. 4 N
D. N

Answer

To determine the number of octahedral sites in a cubic close-packed (CCP) array of N spheres, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure**:
- In a cubic close-packed structure, the spheres are arranged in layers. Each sphere in one layer is surrounded by spheres in the layers above and below it.

2. **Identifying Octahedral Voids**:
- An octahedral void is formed by six atoms. In a CCP structure, there are octahedral voids located at the center of the unit cell and at the edges of the cube.

3. **Counting the Octahedral Sites**:
- In a single unit cell of a cubic close-packed structure, there is 1 octahedral void at the center of the cube and 12 edge-centered octahedral voids.
- Each edge of the cube contributes to one octahedral void, but since each edge is shared by four adjacent unit cells, only 1/4 of each edge-centered void belongs to one unit cell.
- Therefore, the total contribution from the edge-centered octahedral voids is:
\[ \text{Number of edge-centered voids} = 12 \times \frac{1}{4} = 3 \]
- Adding the central octahedral void, the total number of octahedral voids in one unit cell is:
\[ \text{Total octahedral voids} = 1 + 3 = 4 \]

4. **Relating to N Spheres**:
- In a cubic close-packed structure, the number of spheres (N) is related to the number of unit cells. Each unit cell contains 4 spheres in a CCP arrangement.
- Therefore, if we have N spheres, the number of unit cells is \( \frac{N}{4} \).
- Since each unit cell contains 4 octahedral voids, the total number of octahedral voids in N spheres is:
\[ \text{Total octahedral voids} = 4 \times \frac{N}{4} = N \]

### Final Answer:
The number of octahedral sites in a cubic close-packed array of N spheres is **N**.