Calculate the radius of an ion in a body centred cube having edge length a .

Options

A. `a/2 `
B. `(sqrt2a)/(4)`
C. `( sqrt 3a)/(4) `
D. `a`

Answer

To calculate the radius of an ion in a body-centered cubic (BCC) structure with an edge length \( a \), we can follow these steps:

### Step 1: Understand the Structure
In a body-centered cubic unit cell, there are atoms located at each of the eight corners of the cube and one atom at the center of the cube.

### Step 2: Identify the Body Diagonal
The body diagonal of the cube connects two opposite corners of the cube. We can denote the corners of the cube as points A and C, with the center atom at point B.

### Step 3: Apply the Pythagorean Theorem
The body diagonal \( AC \) can be calculated using the Pythagorean theorem. The length of the body diagonal \( AC \) can be expressed in terms of the edge length \( a \):
\[ AC = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3} \]

### Step 4: Relate the Body Diagonal to the Atomic Radius
In a BCC structure, the atoms touch along the body diagonal. The body diagonal \( AC \) consists of the radius of the corner atom (R), the radius of the center atom (R), and the radius of the other corner atom (R). Therefore, we can express this as:
\[ AC = R + 2R + R = 4R \]

### Step 5: Set Up the Equation
Now, we can set the two expressions for \( AC \) equal to each other:
\[ 4R = a\sqrt{3} \]

### Step 6: Solve for the Radius \( R \)
To find the radius \( R \), we rearrange the equation:
\[ R = \frac{a\sqrt{3}}{4} \]

### Conclusion
Thus, the radius of an ion in a body-centered cubic structure with edge length \( a \) is:
\[ R = \frac{a\sqrt{3}}{4} \]