In a face centred cubic unit cell of close packed atoms, the radius of atom (r) is related to the edge length (a) of the unit cell by the expression

Options

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

Answer

To find the relationship between the radius of an atom (r) and the edge length (a) of a face-centered cubic (FCC) unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the FCC Structure**:
- In a face-centered cubic (FCC) unit cell, atoms are located at each of the eight corners of the cube and at the center of each of the six faces.

2. **Identifying the Face Diagonal**:
- The atoms in an FCC unit cell touch each other along the face diagonal of the cube. We need to calculate the length of this face diagonal.

3. **Calculating the Length of the Face Diagonal**:
- Consider one face of the cube. The edge length of the cube is 'a'.
- The face diagonal can be calculated using the Pythagorean theorem:
\[ \text{Face diagonal} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \]

4. **Counting the Atoms Along the Face Diagonal**:
- Along the face diagonal, there are 4 atomic radii (r) in contact:
- One atom at one corner contributes a radius (r).
- One atom at the center of the face contributes another radius (r).
- Another atom at the opposite corner contributes another radius (r).
- Therefore, the total contribution along the face diagonal is:
\[ \text{Total contribution} = 4r \]

5. **Setting Up the Equation**:
- Since the face diagonal is equal to the total contribution of the atomic radii, we can set up the equation:
\[ a\sqrt{2} = 4r \]

6. **Solving for the Radius (r)**:
- Rearranging the equation to find the radius in terms of the edge length:
\[ r = \frac{a\sqrt{2}}{4} \]

### Final Expression:
Thus, the relationship between the radius of the atom (r) and the edge length (a) of the FCC unit cell is given by:
\[ r = \frac{a\sqrt{2}}{4} \]