Find the distance ( in pm) between the body centered atoms and one corner atom in an element `( a = 2.32 "pm")`

Answer

To find the distance between the body-centered atom and one corner atom in a body-centered cubic (BCC) structure, we can follow these steps:

### Step 1: Understand the BCC Structure
In a BCC structure, there are atoms located at the eight corners of the cube and one atom at the center of the cube.

### Step 2: Identify the Body Diagonal
The distance we need to calculate is along the body diagonal of the cube. The body diagonal connects one corner atom to the body-centered atom.

### Step 3: Calculate the Length of the Body Diagonal
The length of the body diagonal (d) of a cube with edge length \(a\) is given by the formula:
\[ d = \sqrt{3} \times a \]

### Step 4: Relate the Body Diagonal to Atomic Radii
In the BCC structure, along the body diagonal, the distance can also be expressed in terms of the atomic radii. The body diagonal consists of one corner atom, the body-centered atom, and another corner atom. The relationship can be expressed as:
\[ d = r + 2r + r = 4r \]
where \(r\) is the radius of the atom.

### Step 5: Set the Two Expressions Equal
From the two expressions for the body diagonal, we have:
\[ \sqrt{3} \times a = 4r \]

### Step 6: Solve for the Radius
Rearranging the equation gives:
\[ r = \frac{\sqrt{3} \times a}{4} \]

### Step 7: Substitute the Given Edge Length
Now, substitute the given edge length \(a = 2.32 \, \text{pm}\):
\[ r = \frac{\sqrt{3} \times 2.32}{4} \]

### Step 8: Calculate the Value of \(r\)
Using \(\sqrt{3} \approx 1.732\):
\[ r = \frac{1.732 \times 2.32}{4} \approx \frac{4.02304}{4} \approx 1.00576 \, \text{pm} \]

### Step 9: Calculate the Distance Between the Body Center Atom and One Corner Atom
The distance between the body-centered atom and one corner atom is simply \(2r\):
\[ \text{Distance} = 2r = 2 \times 1.00576 \approx 2.01152 \, \text{pm} \]

### Final Answer
Thus, the distance between the body-centered atom and one corner atom is approximately:
\[ \text{Distance} \approx 2.01 \, \text{pm} \]