How many number of atoms are completely inside the HCP unit cell ?

Options

A. Exactly 6
B. `gt6`
C. `lt6`
D. 12

Answer

To determine the number of atoms that are completely inside the hexagonal close-packed (HCP) unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure of HCP**:
- The HCP unit cell consists of atoms located at specific positions: corners, face centers, and body centers.

2. **Counting Atoms at the Corners**:
- In an HCP unit cell, there are 12 corner atoms. However, each corner atom is shared among 6 unit cells. Therefore, the contribution of corner atoms to the unit cell is:
\[ \text{Contribution from corner atoms} = 12 \times \frac{1}{6} = 2 \]

3. **Counting Atoms at the Face Centers**:
- There are 2 face-centered atoms in the HCP unit cell. Each face-centered atom is shared between 2 unit cells. Thus, the contribution from face-centered atoms is:
\[ \text{Contribution from face-centered atoms} = 2 \times \frac{1}{2} = 1 \]

4. **Counting Atoms in the Body Center**:
- In the HCP structure, there are 3 atoms located in the body center. These atoms are not shared with any other unit cell, so their contribution is:
\[ \text{Contribution from body-centered atoms} = 3 \times 1 = 3 \]

5. **Calculating the Total Effective Number of Atoms**:
- Now, we can sum up the contributions from all the different positions:
\[ \text{Total effective number of atoms} = 2 \, (\text{from corners}) + 1 \, (\text{from face centers}) + 3 \, (\text{from body centers}) = 6 \]

6. **Conclusion**:
- Therefore, the total number of atoms that are completely inside the HCP unit cell is **6**.

### Final Answer:
The number of atoms completely inside the HCP unit cell is **6**.

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