The relation between atomic radius and edge length 'a' of a body centred cubic unit cell :

Options

A. r=a/2
B. `r=sqrt(a//2)`
C. `r=sqrt3/4a`
D. `r="3a"/2`

Explanation

To find the relation between atomic radius (R) and edge length (a) of a body-centered cubic (BCC) unit cell, we can follow these steps:

### Step 1: Understand the Structure of BCC
A body-centered cubic unit cell has 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
In a BCC unit cell, the body diagonal connects two opposite corners of the cube. The length of the body diagonal can be expressed in terms of the edge length (a) of the cube.

### Step 3: Calculate the Length of the Body Diagonal
The length of the body diagonal (d) can be calculated using the formula:
\[ d = \sqrt{3}a \]

### Step 4: Relate the Body Diagonal to Atomic Radius
Along the body diagonal, there are three atoms: one at each corner and one at the center. The atomic radius (R) of the atoms at the corners contributes to the total length of the body diagonal. The arrangement can be visualized as:
- The distance from one corner atom to the center atom is R.
- The distance from the center atom to the opposite corner atom is also R.

Thus, the total length of the body diagonal in terms of atomic radius is:
\[ \text{Total length} = R + 2R + R = 4R \]

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

### Step 6: Solve for R in Terms of a
Rearranging the equation gives us:
\[ R = \frac{\sqrt{3}}{4}a \]

### Step 7: Express a in Terms of R
Alternatively, we can express the edge length (a) in terms of the atomic radius (R):
\[ a = \frac{4R}{\sqrt{3}} \]

### Final Relation
Thus, the relation between atomic radius (R) and edge length (a) of a body-centered cubic unit cell is:
\[ R = \frac{\sqrt{3}}{4}a \]