the edge langth of the unit cells in terms of the radius of sphere constituting fcc ,bcc and simple cubic unit cells are respectively ………. .

Options

A. `2sqrt(2r),(4r)/(sqrt(3)),2r`
B. `(4r)/(sqrt(3)),2sqrt(2r),2r`
C. `2r,2sqrt(2r),(4r)/(sqrt(3))`
D. `2r,(4r)/(sqrt(3)),2sqrt(2r)`

Explanation

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

### Step-by-Step Solution:

1. **Simple Cubic (SC) Unit Cell:**
- In a simple cubic unit cell, there is one atom at each corner of the cube.
- The edge length \( a \) is equal to twice the radius \( R \) of the sphere because the diameter of the sphere fits along the edge of the cube.
- Therefore, the relationship is:
\[ a = 2R \]

2. **Face-Centered Cubic (FCC) Unit Cell:**
- In an FCC unit cell, there are atoms at each corner and one atom at the center of each face.
- The face diagonal of the cube can be expressed in terms of the radius \( R \) of the spheres. The face diagonal consists of 4 radii (2 spheres) because it passes through the centers of two spheres.
- The length of the face diagonal can also be expressed in terms of the edge length \( a \) using the Pythagorean theorem:
\[ \text{Face diagonal} = a\sqrt{2} \]
- Setting these equal gives:
\[ a\sqrt{2} = 4R \implies a = \frac{4R}{\sqrt{2}} = \frac{4R\sqrt{2}}{2} = 2\sqrt{2}R \]

3. **Body-Centered Cubic (BCC) Unit Cell:**
- In a BCC unit cell, there are atoms at each corner and one atom at the center of the cube.
- The body diagonal of the cube can be expressed in terms of the radius \( R \) of the spheres. The body diagonal consists of 4 radii (2 spheres) because it passes through the centers of two spheres.
- The length of the body diagonal can also be expressed in terms of the edge length \( a \):
\[ \text{Body diagonal} = a\sqrt{3} \]
- Setting these equal gives:
\[ a\sqrt{3} = 4R \implies a = \frac{4R}{\sqrt{3}} \]

### Summary of Relationships:
- For Simple Cubic (SC): \( a = 2R \)
- For Face-Centered Cubic (FCC): \( a = \frac{4R}{\sqrt{2}} \) or \( a = 2\sqrt{2}R \)
- For Body-Centered Cubic (BCC): \( a = \frac{4R}{\sqrt{3}} \)

### Final Answer:
The edge lengths of the unit cells in terms of the radius of the sphere constituting the unit cells are:
- Simple Cubic: \( a = 2R \)
- Face-Centered Cubic: \( a = \frac{4R}{\sqrt{2}} \)
- Body-Centered Cubic: \( a = \frac{4R}{\sqrt{3}} \)