The edge lengths of the unit cells in terms of the radius of spheres constituting fcc, bcc and simple cubic unit cell are respectively

Options

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

Explanation

To find the edge lengths of the unit cells in terms of the radius of the spheres constituting face-centered cubic (FCC), body-centered cubic (BCC), and simple cubic unit cells, we can derive the relationships step by step.

### Step 1: Face-Centered Cubic (FCC)
1. In an FCC unit cell, atoms are located at each corner and the centers of each face.
2. The face diagonal of the cube can be represented as:
\[ \text{Face diagonal} = \sqrt{2} \cdot a \]
where \( a \) is the edge length of the cube.
3. The face diagonal also equals the sum of the diameters of the two atoms along the diagonal:
\[ \text{Face diagonal} = 4r \]
where \( r \) is the radius of the atom.
4. Setting these equal gives:
\[ \sqrt{2} \cdot a = 4r \]
5. Solving for \( a \):
\[ a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r \]

### Step 2: Body-Centered Cubic (BCC)
1. In a BCC unit cell, atoms are located at each corner and one atom is at the center of the cube.
2. The body diagonal of the cube can be represented as:
\[ \text{Body diagonal} = \sqrt{3} \cdot a \]
3. The body diagonal also equals the sum of the diameters of the two corner atoms and the center atom:
\[ \text{Body diagonal} = 4r \]
4. Setting these equal gives:
\[ \sqrt{3} \cdot a = 4r \]
5. Solving for \( a \):
\[ a = \frac{4r}{\sqrt{3}} \]

### Step 3: Simple Cubic (SC)
1. In a simple cubic unit cell, atoms are located only at the corners of the cube.
2. The edge length \( a \) is equal to twice the radius of the atom because there are two radii along the edge:
\[ a = 2r \]

### Summary of Results
- For FCC: \( a = 2\sqrt{2}r \)
- For BCC: \( a = \frac{4r}{\sqrt{3}} \)
- For Simple Cubic: \( a = 2r \)

### Final Answer
The edge lengths of the unit cells in terms of the radius of spheres constituting FCC, BCC, and simple cubic unit cell are respectively:
1. FCC: \( 2\sqrt{2}r \)
2. BCC: \( \frac{4r}{\sqrt{3}} \)
3. Simple Cubic: \( 2r \)