To solve the problem of finding the minimum distance between atoms in a Body-Centered Cubic (BCC) structure with a lattice constant \( a \), we can follow these steps:
### Step-by-Step Solution:
1. **Understanding the BCC Structure**:
- In a BCC lattice, there is one atom located at each of the eight corners of the cube and one atom at the center of the cube.
2. **Visualizing the Cube**:
- Draw a cube to represent the BCC structure. Mark the corners where the atoms are located and the center where the central atom is located.
3. **Identifying Atom Positions**:
- The atom at the center of the cube is at the coordinates (0.5a, 0.5a, 0.5a) and the corner atoms are at (0, 0, 0), (a, 0, 0), (0, a, 0), (0, 0, a), (a, a, 0), (a, 0, a), (0, a, a), and (a, a, a).
4. **Calculating the Diagonal Distance**:
- The distance between two opposite corners of the cube (the body diagonal) can be calculated using the formula for the diagonal of a cube:
\[
d = \sqrt{3}a
\]
5. **Relating the Diagonal to Atomic Radii**:
- In the BCC structure, the body diagonal contains three atomic radii (2 from the corner atoms and 1 from the center atom). Thus, we can express this relationship as:
\[
d = 4r
\]
- Where \( r \) is the atomic radius.
6. **Setting the Equations Equal**:
- We can set the two expressions for the diagonal equal to each other:
\[
4r = \sqrt{3}a
\]
7. **Solving for the Atomic Radius**:
- Rearranging the equation gives us:
\[
r = \frac{\sqrt{3}}{4}a
\]
8. **Finding the Minimum Distance Between Atoms**:
- The minimum distance between two atoms in the BCC structure is twice the atomic radius:
\[
\text{Minimum Distance} = 2r = 2 \left(\frac{\sqrt{3}}{4}a\right) = \frac{\sqrt{3}}{2}a
\]
9. **Conclusion**:
- Therefore, the minimum distance between atoms in a BCC structure is:
\[
\frac{\sqrt{3}}{2}a
\]
### Final Answer:
The minimum distance between atoms in a BCC structure with lattice constant \( a \) is \( \frac{\sqrt{3}}{2}a \).
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