In the body centered cubic unit cell and simple unit cell, the radius of atoms in terms of edge length (a) of the unit cell is respectively:

Options

A. (a) `(a)/(2), (a)/(2sqrt(2))`
B. (b) ` (a)/(sqrt(2)),(a)/(2)`
C. (c) ` (a)/(2sqrt(2)),(a)/(2)`
D. (d) ` (3sqrt(a))/(4),(a)/(2)`

Answer

To find the radius of atoms in terms of the edge length (a) of the unit cell for both the simple cubic unit cell and the body-centered cubic (BCC) unit cell, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Simple Cubic Unit Cell:**
- In a simple cubic unit cell, atoms are located at each of the eight corners of the cube.
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell.
- Therefore, the total number of atoms per unit cell (Z) is:
\[ Z = 8 \times \frac{1}{8} = 1 \]

2. **Finding the Relationship Between Radius and Edge Length in Simple Cubic:**
- In a simple cubic structure, the atoms touch each other along the edge of the cube.
- The edge length \( a \) is equal to twice the radius \( r \) of the atom:
\[ a = 2r \]
- From this, we can express the radius in terms of the edge length:
\[ r = \frac{a}{2} \]

3. **Understanding the Body-Centered Cubic (BCC) Unit Cell:**
- In a BCC unit cell, there are atoms at each of the eight corners and one atom at the center of the cube.
- The total number of atoms per unit cell (Z) is:
\[ Z = 8 \times \frac{1}{8} + 1 = 2 \]

4. **Finding the Relationship Between Radius and Edge Length in BCC:**
- In a BCC structure, the atoms touch along the body diagonal of the cube.
- The length of the body diagonal \( d \) is given by:
\[ d = \sqrt{3}a \]
- Along the body diagonal, there are 4 radii (since there are two atoms, one at each end and one in the center):
\[ d = 4r \]
- Therefore, we can set the two expressions for the body diagonal equal to each other:
\[ \sqrt{3}a = 4r \]
- Solving for \( r \):
\[ r = \frac{\sqrt{3}}{4}a \]

### Final Results:
- For the simple cubic unit cell:
\[ r = \frac{a}{2} \]
- For the body-centered cubic unit cell:
\[ r = \frac{\sqrt{3}}{4}a \]