If `a` is the length of unit cell, then which one is correct relationship ?

Options

A. For simple cubic lattice, Radius of metal atom `=a/2`
B. For bcc lattice, Radius of metal atom `=(sqrt(3)a)/(4)`
C. For fcc lattice, Radius of metal atom `=(a)/(2sqrt(2))`
D. All of these.

Answer

To solve the question regarding the relationships between the length of the unit cell (denoted as `a`) and the radius of the metal atom in different types of crystal lattices, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Simple Cubic Lattice**:
- In a simple cubic lattice, atoms are located at the corners of the cube.
- The effective number of atoms per unit cell (Z) is 1.
- The relationship between the edge length `a` and the radius `r` of the atom is given by:
\[ 2r = a \quad \Rightarrow \quad r = \frac{a}{2} \]
- Therefore, **Option 1** is correct: \( r = \frac{a}{2} \).

2. **Body-Centered Cubic (BCC) Lattice**:
- In a BCC lattice, atoms are located at the corners and one atom is at the body center.
- The effective number of atoms per unit cell (Z) is 2.
- The relationship can be derived using the diagonal of the cube:
- The body diagonal of the cube can be expressed as:
\[ \text{Body diagonal} = \sqrt{3}a \]
- The body diagonal also equals to four times the radius of the atom:
\[ \text{Body diagonal} = 4r \]
- Setting these equal gives:
\[ 4r = \sqrt{3}a \quad \Rightarrow \quad r = \frac{\sqrt{3}}{4}a \]
- Therefore, **Option 2** is correct: \( r = \frac{\sqrt{3}}{4}a \).

3. **Face-Centered Cubic (FCC) Lattice**:
- In an FCC lattice, atoms are located at the corners and the centers of the faces of the cube.
- The effective number of atoms per unit cell (Z) is 4.
- The relationship can be derived using the face diagonal:
- The face diagonal of the cube can be expressed as:
\[ \text{Face diagonal} = \sqrt{2}a \]
- The face diagonal also equals to four times the radius of the atom:
\[ \text{Face diagonal} = 4r \]
- Setting these equal gives:
\[ 4r = \sqrt{2}a \quad \Rightarrow \quad r = \frac{\sqrt{2}}{4}a = \frac{a}{2\sqrt{2}} \]
- Therefore, **Option 3** is correct: \( r = \frac{a}{2\sqrt{2}} \).

4. **Conclusion**:
- Since all three options are correct, the final answer is:
- **Option 4**: All of these statements are correct.