A metal crystallizes in `bcc` lattice. The percent fraction of edge length not covered by atom is

Options

A. `10.4%`
B. `13.4%`
C. `12.4%`
D. `11.4%`

Explanation

To find the percent fraction of edge length not covered by atoms in a body-centered cubic (BCC) lattice, we can follow these steps:

### Step 1: Understand the BCC Structure
In a BCC lattice, atoms are located at the eight corners of a cube and one atom is located at the center of the cube. Each corner atom is shared among eight adjacent unit cells.

### Step 2: Identify the Edge Length and Atomic Radius
Let the edge length of the cube be denoted as \( a \) and the radius of the atom be denoted as \( r \).

### Step 3: Determine the Relationship Between Edge Length and Atomic Radius
In a BCC lattice, the relationship between the edge length \( a \) and the atomic radius \( r \) can be derived from the body diagonal of the cube. The body diagonal passes through the center atom and connects two corner atoms. The length of the body diagonal can be expressed as:
\[ \text{Body diagonal} = \sqrt{3}a \]
Since the body diagonal contains 4 atomic radii (2 from each corner atom and 1 from the center atom), we have:
\[ \sqrt{3}a = 4r \]
From this, we can rearrange to find \( a \):
\[ a = \frac{4r}{\sqrt{3}} \]

### Step 4: Calculate the Length Covered by Atoms on an Edge
Each edge of the cube has two corner atoms, and the length covered by these atoms on one edge is:
\[ \text{Length covered by atoms} = 2r \]

### Step 5: Calculate the Length Not Covered by Atoms
The length of the edge that is not covered by atoms is:
\[ \text{Length not covered} = a - 2r \]

### Step 6: Substitute the Expression for \( a \)
Substituting the expression for \( a \) into the equation:
\[ \text{Length not covered} = \frac{4r}{\sqrt{3}} - 2r \]
To simplify this, we can express \( 2r \) in terms of \( \sqrt{3} \):
\[ \text{Length not covered} = \frac{4r}{\sqrt{3}} - \frac{2r\sqrt{3}}{\sqrt{3}} = \frac{4r - 2r\sqrt{3}}{\sqrt{3}} = \frac{r(4 - 2\sqrt{3})}{\sqrt{3}} \]

### Step 7: Calculate the Percent Fraction Not Covered
To find the percent fraction of the edge length that is not covered by atoms, we divide the length not covered by the total edge length and multiply by 100:
\[ \text{Percent not covered} = \left( \frac{a - 2r}{a} \right) \times 100 \]
Substituting \( a = \frac{4r}{\sqrt{3}} \):
\[ \text{Percent not covered} = \left( \frac{\frac{4r}{\sqrt{3}} - 2r}{\frac{4r}{\sqrt{3}}} \right) \times 100 \]
This simplifies to:
\[ = \left( \frac{4 - 2\sqrt{3}}{4} \right) \times 100 \]

### Step 8: Calculate the Numerical Value
Calculating \( 4 - 2\sqrt{3} \):
- \( \sqrt{3} \approx 1.732 \)
- \( 2\sqrt{3} \approx 3.464 \)
- \( 4 - 3.464 \approx 0.536 \)

Now substituting back:
\[ \text{Percent not covered} \approx \left( \frac{0.536}{4} \right) \times 100 \approx 13.4\% \]

### Final Answer
Thus, the percent fraction of edge length not covered by atoms in a BCC lattice is approximately **13.4%**.