In face -centered cubic unit cell, edge length is

Options

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

Answer

To find 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 an FCC unit cell, atoms are located at each of the eight corners and at the centers of each of the six faces of the cube.

2. **Defining Variables**:
- Let the edge length of the unit cell be denoted as \( A \).
- Let the radius of the atoms be denoted as \( R \).

3. **Identifying the Face Diagonal**:
- The face diagonal of the FCC unit cell can be visualized as the diagonal that connects two opposite corners of a face of the cube.
- According to the Pythagorean theorem, the length of the face diagonal (D) can be calculated as:
\[ D = \sqrt{A^2 + A^2} = \sqrt{2A^2} = A\sqrt{2} \]

4. **Relating the Face Diagonal to Atomic Radius**:
- The face diagonal in terms of atomic radius can be expressed as the sum of the diameters of the atoms along the diagonal.
- In the FCC structure, there are four atomic radii along the face diagonal (two atoms from the corners and two from the face centers):
\[ D = 4R \]

5. **Equating the Two Expressions for the Diagonal**:
- We can now set the two expressions for the face diagonal equal to each other:
\[ A\sqrt{2} = 4R \]

6. **Solving for Edge Length (A)**:
- To find the edge length \( A \), we rearrange the equation:
\[ A = \frac{4R}{\sqrt{2}} = \frac{4R \sqrt{2}}{2} = 2\sqrt{2}R \]

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