In the structure of diamond, carbon atoms appear at
a.`0, 0, 0`, and `(1)/(2), (1)/(2), (1)/(2)`
b.`(1)/(4), (1)/(4), (1)/4`, and `(1)/(2), (1)/(2), (1)/(2)`
c.`0, 0, 0`, and `(1)/(4), (1)/(4), (1)/(4)`
d.`0, 0, 0`, and `(3)/(4), (3)/(4), (3)/(4)`

Options

A. `0, 0, 0`, and `(1)/(2), (1)/(2), (1)/(2)`
B. `(1)/(4), (1)/(4), (1)/4`, and `(1)/(2), (1)/(2), (1)/(2)`
C. `0, 0, 0`, and `(1)/(4), (1)/(4), (1)/(4)`
D. `0, 0, 0`, and `(3)/(4), (3)/(4), (3)/(4)`

Explanation

To determine the positions of carbon atoms in the structure of diamond, we need to analyze the crystal structure of diamond, which is a type of face-centered cubic (FCC) lattice with additional carbon atoms located at tetrahedral sites.

### Step-by-Step Solution:

1. **Understanding the Diamond Structure**:
- Diamond has a face-centered cubic (FCC) structure.
- In addition to the atoms at the corners and face centers of the cube, diamond has carbon atoms located at tetrahedral positions.

2. **Identifying the Positions**:
- The carbon atoms in diamond occupy specific positions:
- One carbon atom is located at the origin, which is represented as (0, 0, 0).
- Another carbon atom is located at the position (1/2, 1/2, 1/2), which is the body center of the cubic unit cell.

3. **Evaluating the Options**:
- **Option a**: (0, 0, 0) and (1/2, 1/2, 1/2) – This matches our identified positions.
- **Option b**: (1/4, 1/4, 1/4) and (1/2, 1/2, 1/2) – This does not match as (1/4, 1/4, 1/4) is not a position for carbon in diamond.
- **Option c**: (0, 0, 0) and (1/4, 1/4, 1/4) – This does not match as (1/4, 1/4, 1/4) is not a position for carbon in diamond.
- **Option d**: (0, 0, 0) and (3/4, 3/4, 3/4) – This does not match as (3/4, 3/4, 3/4) is also not a position for carbon in diamond.

4. **Conclusion**:
- The correct positions of carbon atoms in the diamond structure are given by option **a**: (0, 0, 0) and (1/2, 1/2, 1/2).

### Final Answer:
**Option a: (0, 0, 0) and (1/2, 1/2, 1/2)**