The lattice parameter for a simple cubic unit cell of atomic radius 'r' is

Options

A. `2r`
B. `(4r)/(sqrt3)`
C. `sqrt8r`
D. `(sqrt3)/(4)r`

Answer

To find the lattice parameter (edge length) for a simple cubic unit cell with atomic radius 'r', follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure of Simple Cubic Unit Cell**:
- In a simple cubic unit cell, atoms are located at each of the eight corners of the cube.

2. **Identify the Atomic Radius**:
- The atomic radius is denoted by 'r'. This is the distance from the center of an atom to its outer edge.

3. **Determine the Nearest Neighbor Distance**:
- In a simple cubic structure, the nearest neighbors are the atoms located at the corners of the cube. The distance between the centers of two adjacent corner atoms is the nearest neighbor distance, denoted as 'd'.

4. **Relate Nearest Neighbor Distance to Edge Length**:
- The nearest neighbor distance 'd' in a simple cubic unit cell is equal to the edge length 'a' of the unit cell. Therefore, we can write:
\[ d = a \]

5. **Express Nearest Neighbor Distance in Terms of Atomic Radius**:
- Since the nearest neighbor distance 'd' is the distance between the centers of two adjacent atoms, it can also be expressed as:
\[ d = 2r \]
- This is because the distance from the center of one atom to the center of the adjacent atom is the sum of their radii (r + r = 2r).

6. **Set the Two Expressions Equal**:
- From the previous steps, we have:
\[ a = d = 2r \]

7. **Conclusion**:
- Therefore, the lattice parameter (edge length) 'a' for a simple cubic unit cell in terms of the atomic radius 'r' is:
\[ a = 2r \]

### Final Answer:
The lattice parameter for a simple cubic unit cell of atomic radius 'r' is \( a = 2r \).

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