The portion of edge length not occupied by atoms for simple cubic, fcc and bcc are respectively (a is edge length)

Options

A. `0, a(1 - (sqrt3)/2) , a(1 - 1/(sqrt2))`
B. `a(1 - (sqrt3)/(2)) , 0 , a(2 - 1/(sqrt(2)))`
C. `0, a(1 - 1/(sqrt(2)) , a (1 - (sqrt3)/(2))`
D. `a , 2sqrt(2)a , (sqrt3)/2 a`

Explanation

To find the portion of edge length not occupied by atoms for simple cubic (SC), face-centered cubic (FCC), and body-centered cubic (BCC) structures, we will follow these steps:

### Step 1: Simple Cubic (SC)

1. **Understanding the Structure**: In a simple cubic structure, there is one atom at each corner of the cube. The edge length (a) is related to the radius (R) of the atom.

2. **Relation**: The edge length \( a \) is equal to \( 2R \) because the atoms at the corners touch each other along the edge.

3. **Occupied Portion**: Since the edge length is fully occupied by the two radii (one from each corner atom), the portion occupied by atoms is \( 2R \).

4. **Unoccupied Portion**: The unoccupied portion of the edge length is:
\[ \text{Unoccupied Portion} = a - \text{Occupied Portion} = a - 2R = 0 \]

### Step 2: Face-Centered Cubic (FCC)

1. **Understanding the Structure**: In an FCC structure, there are atoms at each corner and one atom at the center of each face.

2. **Relation**: The diagonal of the face of the cube can be expressed as:
\[ a\sqrt{2} = 4R \]
This means that the edge length \( a \) is related to the radius \( R \) by:
\[ a = \frac{4R}{\sqrt{2}} = 2\sqrt{2}R \]

3. **Occupied Portion**: The distance between the centers of two atoms along the edge is \( R + R = 2R \).

4. **Unoccupied Portion**: The unoccupied portion of the edge length is:
\[ \text{Unoccupied Portion} = a - 2R = 2\sqrt{2}R - 2R \]
Simplifying this:
\[ = 2R(\sqrt{2} - 1) \]

### Step 3: Body-Centered Cubic (BCC)

1. **Understanding the Structure**: In a BCC structure, there are atoms at each corner and one atom at the center of the cube.

2. **Relation**: The body diagonal of the cube can be expressed as:
\[ a\sqrt{3} = 4R \]
This means that the edge length \( a \) is related to the radius \( R \) by:
\[ a = \frac{4R}{\sqrt{3}} = \frac{4R\sqrt{3}}{3} \]

3. **Occupied Portion**: The distance between the centers of two corner atoms along the body diagonal is \( R + R + R = 3R \).

4. **Unoccupied Portion**: The unoccupied portion of the edge length is:
\[ \text{Unoccupied Portion} = a - 3R = \frac{4R\sqrt{3}}{3} - 3R \]
Simplifying this:
\[ = R\left(\frac{4\sqrt{3}}{3} - 3\right) \]

### Final Results

- Simple Cubic: \( 0 \)
- Face-Centered Cubic: \( 2R(\sqrt{2} - 1) \)
- Body-Centered Cubic: \( R\left(\frac{4\sqrt{3}}{3} - 3\right) \)