The radius of the metal atom can be expressed in terms of the length of a unit cell is `:`

Options

A. it is `a//2` for simple cubic lattice
B. it is `(sqrt(3)a//4)` for b.c.c. lattice
C. it is `(a//2sqrt(2))` for F.C.C. lattice
D. All of the above

Answer

To find the radius of a metal atom in terms of the length of a unit cell for different types of crystal lattices, we can analyze each structure: Simple Cubic, Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC).

### Step-by-Step Solution:

1. **Simple Cubic Lattice:**
- In a simple cubic lattice, each corner of the cube has one atom.
- The edge length of the unit cell is denoted as \( a \).
- The atoms at the corners touch each other along the edge of the cube.
- Therefore, the relationship between the radius \( r \) of the atom and the edge length \( a \) is:
\[ a = r + r = 2r \]
- Rearranging gives:
\[ r = \frac{a}{2} \]

2. **Body-Centered Cubic (BCC) Lattice:**
- In a BCC lattice, there is one atom at each corner and one atom in the center of the cube.
- The body diagonal of the cube contains 4 atomic radii (from corner to corner through the center atom).
- The length of the body diagonal can be expressed in terms of the edge length \( a \) as:
\[ \text{Body diagonal} = \sqrt{3}a \]
- Thus, we have:
\[ 4r = \sqrt{3}a \]
- Rearranging gives:
\[ r = \frac{\sqrt{3}a}{4} \]

3. **Face-Centered Cubic (FCC) Lattice:**
- In an FCC lattice, there are atoms at each corner and one atom at the center of each face.
- The face diagonal of the cube contains 4 atomic radii.
- The length of the face diagonal can be expressed in terms of the edge length \( a \) as:
\[ \text{Face diagonal} = \sqrt{2}a \]
- Thus, we have:
\[ 4r = \sqrt{2}a \]
- Rearranging gives:
\[ r = \frac{\sqrt{2}a}{4} = \frac{a}{2\sqrt{2}} \]

### Summary of Results:
- For Simple Cubic: \( r = \frac{a}{2} \)
- For BCC: \( r = \frac{\sqrt{3}a}{4} \)
- For FCC: \( r = \frac{a}{2\sqrt{2}} \)

### Conclusion:
All three expressions for the radius \( r \) of the metal atom in terms of the unit cell length \( a \) are valid for their respective lattice types. Therefore, the correct answer is that all options are correct.