In an atomic bcc lattie what fraction of edge is not covered by atoms?

Options

A. 0.32
B. 0.16
C. 0.134
D. 0.268

Answer

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

### Step-by-Step Solution:

1. **Understanding the BCC Structure**:
- In a BCC lattice, atoms are located at the eight corners of the cube and one atom is located at the center of the cube.

2. **Identifying the Edge Length**:
- Let the edge length of the unit cell be denoted as \( a \).

3. **Identifying the Radius of the Atoms**:
- Let the radius of the atoms be denoted as \( r \).

4. **Calculating the Covered Length on the Edge**:
- On an edge of the BCC unit cell, two corner atoms are present. Each corner atom contributes a radius \( r \) to the edge length.
- Therefore, the total length covered by the two corner atoms on one edge is \( 2r \).

5. **Calculating the Uncovered Length**:
- The length of the edge that is not covered by atoms can be calculated as:
\[ \text{Uncovered Length} = a - 2r \]

6. **Finding the Relationship Between \( r \) and \( a \)**:
- To find the relationship between the radius \( r \) and the edge length \( a \), we can use the body diagonal of the cube.
- The body diagonal of the cube can be expressed as:
\[ \text{Body Diagonal} = \sqrt{3}a \]
- This body diagonal consists of one atom at each corner and one atom at the center, which gives:
\[ \text{Body Diagonal} = 4r \]
- Therefore, we can set up the equation:
\[ 4r = \sqrt{3}a \]
- From this, we can express \( r \) in terms of \( a \):
\[ r = \frac{\sqrt{3}}{4}a \]

7. **Substituting \( r \) Back into the Uncovered Length**:
- Substitute \( r \) into the uncovered length equation:
\[ \text{Uncovered Length} = a - 2\left(\frac{\sqrt{3}}{4}a\right) = a - \frac{\sqrt{3}}{2}a = a\left(1 - \frac{\sqrt{3}}{2}\right) \]

8. **Calculating the Fraction of the Edge Not Covered**:
- The fraction of the edge that is not covered by atoms is given by:
\[ \text{Fraction Not Covered} = \frac{\text{Uncovered Length}}{a} = 1 - \frac{\sqrt{3}}{2} \]
- Evaluating this gives:
\[ \text{Fraction Not Covered} \approx 1 - 0.866 \approx 0.134 \]

9. **Final Result**:
- Therefore, the fraction of the edge that is not covered by atoms in a BCC lattice is approximately \( 0.134 \).