What is the height of an HCP unit cell ?

Options

A. `(sqrt3)/(2)a`
B. `sqrt((3)/(2))a`
C. `sqrt((2)/(3)) xx 2a`
D. `(sqrt2)/(3) a`

Answer

To find the height of an HCP (Hexagonal Close Packed) unit cell, we can follow these steps:

### Step 1: Understand the Structure of HCP
The HCP structure consists of layers of atoms arranged in a hexagonal pattern. Each unit cell has a specific height and base dimensions.

### Step 2: Visualize the Layers
In HCP, there are two types of layers:
- **Layer 1**: The first layer of atoms.
- **Layer 2**: The second layer of atoms that sits in the depressions of the first layer.

### Step 3: Identify the Parameters
Let’s denote:
- The side length of the hexagon as \( a \).
- The height of the unit cell as \( H \).

### Step 4: Determine the Height Using Geometry
The atoms in the HCP structure can be visualized as forming a right triangle when looking at the arrangement of the layers. The height \( H \) can be derived from the geometry of the triangle formed by the centers of the atoms in the layers.

1. The distance between the centers of two atoms in the same layer is \( 2a \) (where \( a \) is the radius of the atom).
2. The vertical distance (height) between the two layers can be calculated using the Pythagorean theorem.

### Step 5: Apply the Pythagorean Theorem
We can set up the equation based on the right triangle formed:
\[ H^2 + \left(\frac{2a}{\sqrt{3}}\right)^2 = (2a)^2 \]
Here, \( \frac{2a}{\sqrt{3}} \) is the horizontal distance between the centers of the atoms in adjacent layers.

### Step 6: Solve for \( H \)
Substituting the values into the equation:
\[ H^2 + \frac{4a^2}{3} = 4a^2 \]
Rearranging gives:
\[ H^2 = 4a^2 - \frac{4a^2}{3} \]
\[ H^2 = \frac{12a^2}{3} - \frac{4a^2}{3} = \frac{8a^2}{3} \]
Taking the square root:
\[ H = \sqrt{\frac{8a^2}{3}} = \frac{2a}{\sqrt{3}} \]

### Conclusion
Thus, the height of an HCP unit cell is:
\[ H = \frac{2a}{\sqrt{3}} \]