In a face- centred cubic unit cell, the edge length is

Options

A. `4/(sqrt3) r`
B. `4/(sqrt2) r`
C. `2 r`
D. `(sqrt3)/2 r`

Answer

To determine the edge length of a face-centered cubic (FCC) unit cell in terms of the atomic radius (R), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the FCC Structure**:
- In a face-centered cubic unit cell, there are atoms located at each of the eight corners and one atom at the center of each of the six faces.
- Each corner atom contributes 1/8 of an atom to the unit cell, and each face-centered atom contributes 1/2 of an atom.

2. **Calculating Total Number of Atoms in FCC**:
- Total contribution from corner atoms: \(8 \times \frac{1}{8} = 1\)
- Total contribution from face-centered atoms: \(6 \times \frac{1}{2} = 3\)
- Therefore, the total number of atoms in an FCC unit cell is \(1 + 3 = 4\).

3. **Identifying the Face Diagonal**:
- The face diagonal of the cube can be represented as the line that connects two opposite corners of a face of the cube.
- The length of the face diagonal (d) can be expressed in terms of the edge length (A) of the cube using the Pythagorean theorem:
\[ d = \sqrt{A^2 + A^2} = A\sqrt{2} \]

4. **Relating Face Diagonal to Atomic Radius**:
- In the FCC structure, the face diagonal is equal to the sum of the diameters of the atoms that touch along this diagonal.
- Along the face diagonal, there are 4 atomic radii (since there are two corner atoms and one face-centered atom):
\[ d = 4R \]

5. **Setting Up the Equation**:
- We can set the two expressions for the face diagonal equal to each other:
\[ A\sqrt{2} = 4R \]

6. **Solving for Edge Length (A)**:
- Rearranging the equation to solve for A gives:
\[ A = \frac{4R}{\sqrt{2}} = 2\sqrt{2}R \]

### Final Answer:
The edge length \(A\) of a face-centered cubic unit cell in terms of the atomic radius \(R\) is:
\[ A = 2\sqrt{2}R \]