To find the minimum distance between atoms in a Body-Centered Cubic (BCC) structure with a lattice constant \( a \), we can follow these steps:
### Step 1: Understand the BCC Structure
In a BCC structure, there are atoms located at each of the eight corners of a cube and one atom at the center of the cube. The lattice constant \( a \) represents the edge length of the cube.
### Step 2: Identify the Atoms and Their Arrangement
The atoms at the corners are positioned at coordinates such as (0,0,0), (0,0,a), (0,a,0), (a,0,0), etc. The atom in the center is located at (a/2, a/2, a/2).
### Step 3: Calculate the Distance Between Atoms
To find the minimum distance between atoms, we need to consider the distance between the corner atoms and the center atom. The distance between two corner atoms can be calculated using the diagonal of the cube.
### Step 4: Use the Diagonal Distance Formula
The distance \( d \) between two corner atoms that are diagonally opposite in the cube can be calculated using the formula:
\[
d = \sqrt{(a)^2 + (a)^2 + (a)^2} = \sqrt{3a^2} = a\sqrt{3}
\]
### Step 5: Relate the Distance to Atomic Radius
In a BCC structure, the distance between the center atom and a corner atom is equal to \( 4r \), where \( r \) is the atomic radius. Thus, we can set up the equation:
\[
4r = a\sqrt{3}
\]
### Step 6: Solve for the Atomic Radius
From the equation above, we can solve for \( r \):
\[
r = \frac{a\sqrt{3}}{4}
\]
### Step 7: Find the Minimum Distance Between Atoms
The minimum distance between two atoms (which is the distance between two touching atoms) is given by:
\[
\text{Minimum Distance} = 2r
\]
Substituting the value of \( r \):
\[
\text{Minimum Distance} = 2 \times \frac{a\sqrt{3}}{4} = \frac{a\sqrt{3}}{2}
\]
### Conclusion
Thus, the minimum distance between atoms in a BCC structure with lattice constant \( a \) is:
\[
\text{Minimum Distance} = \frac{a\sqrt{3}}{2}
\]
