A metal (M), shows ABAB arrangement of atoms in solid state, then what is the relatin between radius of atom (r) and edge length (a) and height (c) of HCP unit cell

Options

A. `a = 2R`
B. `c = (sqrt((2)/(3)))4r`
C. `c = (sqrt((3)/(2)))4r`
D. `a = (2sqrt2)r`

Explanation

To solve the problem, we need to establish the relationships between the radius of the atom (r), the edge length (a), and the height (c) of the hexagonal close-packed (HCP) unit cell.

### Step-by-Step Solution:

1. **Identify the Structure**:
The problem states that the metal (M) shows an ABAB arrangement of atoms in the solid state. This arrangement is characteristic of a hexagonal close-packed (HCP) structure.

2. **Relationship Between Edge Length and Radius**:
In an HCP unit cell, the atoms are arranged in such a way that the edge length (a) is related to the radius (r) of the atoms.
- The relationship is given by:
\[ a = 2r \]
This is because in the HCP structure, the distance between the centers of two adjacent atoms along the edge of the unit cell is equal to twice the radius of the atom.

3. **Height of the HCP Unit Cell**:
The height (c) of the HCP unit cell can be derived from its geometry. The relationship between the height (c) and the edge length (a) is given by:
\[ c = \frac{\sqrt{2}}{3} \times 2a \]
Simplifying this gives:
\[ c = \frac{2\sqrt{2}}{3} a \]

4. **Substituting Edge Length in Terms of Radius**:
Now, we can substitute the expression for edge length (a) in terms of radius (r) into the equation for height (c):
\[ c = \frac{2\sqrt{2}}{3} \times 2r \]
Simplifying this gives:
\[ c = \frac{4\sqrt{2}}{3} r \]

5. **Final Relationships**:
We have established two important relationships:
- The relationship between the radius and edge length:
\[ a = 2r \]
- The relationship between the height and radius:
\[ c = \frac{4\sqrt{2}}{3} r \]

### Summary of Relationships:
- **Edge Length and Radius**: \( a = 2r \)
- **Height and Radius**: \( c = \frac{4\sqrt{2}}{3} r \)