The interatomic distance for bcc crystal is

Options

A. a
B. `a/2`
C. `(sqrt3a)/2`
D. `(2a)/sqrt3`

Answer

To find the interatomic distance for a body-centered cubic (BCC) crystal structure, we can follow these steps:

### Step-by-Step Solution:

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

2. **Identify the Body Diagonal**: The atoms in a BCC structure touch along the body diagonal of the cube. The body diagonal connects two opposite corners of the cube through the center atom.

3. **Calculate the Length of the Body Diagonal**: The length of the body diagonal (D) of a cube with edge length 'a' can be calculated using the formula:
\[ D = a \sqrt{3} \]

4. **Determine the Relationship Between the Radius and the Body Diagonal**: In the BCC structure, the body diagonal is equal to the sum of the diameters of the atoms along that diagonal. Since there are two corner atoms and one center atom, the total distance covered by the atoms along the body diagonal is:
\[ D = 4r \]
where 'r' is the radius of the atom.

5. **Set the Equations Equal**: We can set the two expressions for the body diagonal equal to each other:
\[ a \sqrt{3} = 4r \]

6. **Solve for the Edge Length 'a' in Terms of 'r'**: Rearranging the equation gives:
\[ a = \frac{4r}{\sqrt{3}} \]

7. **Calculate the Interatomic Distance**: The interatomic distance (D) is defined as the distance between the centers of two adjacent atoms. Since the atoms touch along the body diagonal, we can express the interatomic distance as:
\[ D = 2r \]

8. **Substituting for 'r'**: From our earlier equation, we can express 'r' in terms of 'a':
\[ r = \frac{a \sqrt{3}}{4} \]
Therefore, substituting this back into the equation for interatomic distance gives:
\[ D = 2 \left(\frac{a \sqrt{3}}{4}\right) = \frac{a \sqrt{3}}{2} \]

### Final Result:
The interatomic distance for a BCC crystal structure is:
\[ D = \frac{a \sqrt{3}}{2} \]