What is the nearest distance between two different layers in ABAB arrangement (`a = 2 xx` radius of the particle)

Options

A. `sqrt((8)/(3))a`
B. `sqrt((4)/(3))a`
C. `(1)/(sqrt6) a`
D. `sqrt((2)/(3))a`

Answer

To find the nearest distance between two different layers in an ABAB arrangement (which corresponds to hexagonal close packing, HCP), we can follow these steps:

### Step-by-Step Solution

1. **Understand the Arrangement**: In an ABAB arrangement, we have alternating layers of two types of particles (A and B). This is characteristic of hexagonal close packing (HCP).

2. **Identify the Parameter**: The parameter given is \( a = 2r \), where \( r \) is the radius of the particles. This means that the distance between the centers of two adjacent particles in the same layer (A or B) is \( 2r \).

3. **Determine the Nearest Distance**: The nearest distance between two different layers (one A layer and one B layer) can be derived from the geometry of the packing. For HCP, the nearest distance \( D \) between two different layers is given by the formula:
\[ D = \sqrt{\frac{2}{3}} A \]
where \( A \) is the distance between the centers of two adjacent particles in the same layer.

4. **Substitute the Value of \( A \)**: Since \( A = 2r \), we can substitute this into the formula:
\[ D = \sqrt{\frac{2}{3}} (2r) = \frac{2\sqrt{2}}{\sqrt{3}} r \]

5. **Final Expression**: Thus, the nearest distance between two different layers in an ABAB arrangement is:
\[ D = \frac{2\sqrt{2}}{\sqrt{3}} r \]

### Summary of the Solution
The nearest distance between two different layers in an ABAB arrangement (HCP) is given by:
\[ D = \frac{2\sqrt{2}}{\sqrt{3}} r \]