For bcc metallic unit cell, the edge length (a) and radius (r) of aotm are related as : `a = (4)/(sqrt3)r`. True or false?

Answer

To determine whether the relation \( a = \frac{4}{\sqrt{3}} r \) is true or false for a body-centered cubic (BCC) metallic unit cell, we can follow these steps:

### Step 1: Understand the BCC Structure
In a BCC unit cell, there is one atom 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 atoms that touch each other are located along the body diagonal of the cube. The body diagonal connects two opposite corners of the cube and passes through the center atom.

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

### Step 4: Relate the Body Diagonal to Atomic Radius
In the BCC structure, the body diagonal is equal to the sum of the diameters of the two corner atoms and the diameter of the center atom. Since the radius of each atom is \( r \), the diameter is \( 2r \). Therefore, along the body diagonal, we have:
\[ d = 4r \]
This is because there are two corner atoms (each contributing a diameter of \( 2r \)) and one center atom (also contributing a diameter of \( 2r \)):
\[ d = 2r + 2r = 4r \]

### 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:
\[ a\sqrt{3} = 4r \]

### Step 6: Solve for Edge Length \( a \)
To find the relationship between \( a \) and \( r \), we can rearrange the equation:
\[ a = \frac{4r}{\sqrt{3}} \]

### Conclusion
The derived relationship \( a = \frac{4}{\sqrt{3}} r \) is indeed true.

### Final Answer
**True**