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=sqrt(3)/4 a`
D. `r=(3a)/2`

Answer

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

### Step 1: Understand the Structure of BCC
In a body-centered cubic unit cell, atoms are located at each of the eight corners of the cube and one atom is located 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 and passes through the center atom. The length of the body diagonal can be expressed in terms of the atomic radius.

### Step 3: Calculate the Length of the Body Diagonal
In a BCC unit cell, the body diagonal (d) can be represented as:
\[ d = 4r \]
where \( r \) is the atomic radius. This is because the body diagonal consists of the radius from one corner atom to the center atom and then from the center atom to the opposite corner atom.

### Step 4: Relate the Body Diagonal to Edge Length
The body diagonal can also be calculated using the edge length (a) of the cube. The relationship is given by the Pythagorean theorem:
\[ d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3} \]

### Step 5: Set the Two Expressions for the Body Diagonal Equal
Now, we can set the two expressions for the body diagonal equal to each other:
\[ 4r = a\sqrt{3} \]

### Step 6: Solve for the Atomic Radius
To find the relationship between the atomic radius and the edge length, we can rearrange the equation:
\[ r = \frac{a\sqrt{3}}{4} \]

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