In a close packed array of N spheres, the number of tetrahedral holes are

Options

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

Answer

To determine the number of tetrahedral holes in a close-packed array of N spheres, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Close-Packed Structures**:
- In a close-packed structure, spheres are arranged in a way that maximizes the packing efficiency. The two common types of close-packed structures are Face-Centered Cubic (FCC) and Hexagonal Close-Packed (HCP).

2. **Identifying Tetrahedral and Octahedral Holes**:
- In any close-packed arrangement, there are two types of voids: tetrahedral and octahedral.
- The number of octahedral voids is equal to the number of spheres (N).
- The number of tetrahedral voids is related to the number of octahedral voids.

3. **Relationship Between Tetrahedral and Octahedral Holes**:
- For every octahedral void, there are two tetrahedral voids. Therefore, the number of tetrahedral holes can be calculated as:
\[ \text{Number of Tetrahedral Holes} = 2 \times \text{Number of Octahedral Holes} \]

4. **Calculating the Number of Tetrahedral Holes**:
- Since the number of octahedral holes is equal to N (the number of spheres), we can substitute this into the equation:
\[ \text{Number of Tetrahedral Holes} = 2 \times N \]

5. **Conclusion**:
- Thus, the number of tetrahedral holes in a close-packed array of N spheres is:
\[ \text{Number of Tetrahedral Holes} = 2N \]

### Final Answer:
The number of tetrahedral holes in a close-packed array of N spheres is \( 2N \).

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