The number of atoms present in a hexagonal close-packed unit cell is:

Options

A. 4
B. 6
C. 8
D. 12

Answer

To find the number of atoms present in a hexagonal close-packed (HCP) unit cell, we can follow these steps:

### Step 1: Understand the Structure of HCP
The hexagonal close-packed structure consists of layers of atoms arranged in an AB-AB stacking pattern. This means that the first layer (A) is followed by the second layer (B), and then the first layer (A) is repeated.

### Step 2: Identify the Positions of Atoms
In an HCP unit cell:
- There are atoms located at the corners of the hexagonal base.
- There are atoms located at the face centers of the hexagonal faces.
- There are atoms located in the body center of the unit cell.

### Step 3: Count the Corner Atoms
In the HCP unit cell, there are 12 corner atoms. Each corner atom is shared by 6 adjacent unit cells, so the contribution of each corner atom to the unit cell is \( \frac{1}{6} \).

- Total contribution from corner atoms:
\[ \text{Contribution from corners} = 12 \times \frac{1}{6} = 2 \]

### Step 4: Count the Face-Centered Atoms
There are 2 face-centered atoms in the HCP unit cell (one on the top face and one on the bottom face). Each face-centered atom is shared by 2 unit cells, so the contribution of each face-centered atom to the unit cell is \( \frac{1}{2} \).

- Total contribution from face-centered atoms:
\[ \text{Contribution from face centers} = 2 \times \frac{1}{2} = 1 \]

### Step 5: Count the Body-Centered Atoms
In the HCP unit cell, there are 3 body-centered atoms. These atoms are not shared with any other unit cell, so their contribution is 1 each.

- Total contribution from body-centered atoms:
\[ \text{Contribution from body centers} = 3 \times 1 = 3 \]

### Step 6: Calculate the Total Number of Atoms
Now, we can sum up the contributions from the corner atoms, face-centered atoms, and body-centered atoms to find the total number of atoms in the HCP unit cell.

\[ \text{Total number of atoms} = \text{Contribution from corners} + \text{Contribution from face centers} + \text{Contribution from body centers} \]
\[ \text{Total number of atoms} = 2 + 1 + 3 = 6 \]

### Conclusion
Thus, the total number of atoms present in a hexagonal close-packed unit cell is **6**.

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