The shortest distance between `I^(st) and V^(th)` layer of HCP arrangement is:

Options

A. `8 sqrt((2)/(3)) r`
B. `4sqrt((3)/(2))r`
C. `16(sqrt12)/(3)r`
D. `8sqrt((3)/(2))r`

Answer

To find the shortest distance between the first and the fifth layer of a Hexagonal Close Packing (HCP) arrangement, we can follow these steps:

### Step 1: Understand the HCP Structure
In an HCP structure, the layers are arranged in a specific manner. The first layer (A) is followed by the second layer (B), which is then followed by the third layer (A again), the fourth layer (B), and the fifth layer (A). The arrangement alternates between two types of layers.

### Step 2: Identify the Distance Between Layers
The distance we are looking for is the vertical distance between the first layer (A) and the fifth layer (A). In HCP, the distance between two consecutive layers can be determined by the height of the tetrahedron formed by the atoms in the layers.

### Step 3: Calculate the Height of a Tetrahedron
The height (h) of a tetrahedron can be calculated using the formula:
\[ h = \frac{2}{\sqrt{3}} R \]
where \( R \) is the radius of the atoms in the structure.

### Step 4: Calculate the Total Distance
Since the distance between the first layer and the fifth layer involves moving through four heights (from the first layer to the second, second to the third, third to the fourth, and fourth to the fifth), we can express the total distance (D) as:
\[ D = 4h \]
Substituting the height of the tetrahedron:
\[ D = 4 \left( \frac{2}{\sqrt{3}} R \right) = \frac{8}{\sqrt{3}} R \]

### Step 5: Final Expression
Thus, the shortest distance between the first and the fifth layer of the HCP arrangement is:
\[ D = \frac{8R}{\sqrt{3}} \]

### Summary
The shortest distance between the first and fifth layers of an HCP arrangement is given by:
\[ D = \frac{8R}{\sqrt{3}} \]

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