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

Options

A. 4
B. 3
C. 5
D. 6

Answer

To determine 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 has an arrangement of atoms that can be described as A-B-A-B layers. This means that the atoms in one layer are directly above the gaps of the layer below.

**Hint:** Remember that HCP has an alternating layer arrangement.

### Step 2: Count the Corner Atoms
In a hexagonal unit cell, there are 12 corners. Each corner atom is shared among six adjacent unit cells. Therefore, the contribution of each corner atom to the unit cell is \( \frac{1}{6} \).

- Total contribution from corners = \( 12 \times \frac{1}{6} = 2 \) atoms.

**Hint:** Each corner atom contributes only a fraction to the unit cell because it is shared with other unit cells.

### Step 3: Count the Face-Centered Atoms
In the hexagonal unit cell, there are 2 face-centered atoms (one on each of the two hexagonal faces). Each face-centered atom is shared between two unit cells, so the contribution from each face-centered atom is \( \frac{1}{2} \).

- Total contribution from face-centered atoms = \( 2 \times \frac{1}{2} = 1 \) atom.

**Hint:** Face-centered atoms contribute half because they are shared with another unit cell.

### Step 4: Count the Body-Centered Atoms
In the hexagonal close-packed structure, there are 3 atoms located in the body center of the unit cell. These atoms are not shared with any other unit cell, so their contribution is 1 each.

- Total contribution from body-centered atoms = \( 3 \times 1 = 3 \) atoms.

**Hint:** Body-centered atoms are fully contained within the unit cell.

### Step 5: Calculate the Total Number of Atoms
Now, we can sum up the contributions from the corners, face-centered, and body-centered atoms:

- Total number of atoms = Contribution from corners + Contribution from face-centered + Contribution from body-centered
- Total number of atoms = \( 2 + 1 + 3 = 6 \).

### Final Answer
The total number of atoms present in a hexagonal close-packed unit cell is **6**.

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