The largest diagonal in a cubic crystal of edge length a is

Options

A. `sqrt(3a)`
B. `sqrt2a`
C. `sqrt3a`
D. `sqrt(2a)`

Answer

To find the largest diagonal in a cubic crystal of edge length \( a \), we need to consider the two types of diagonals present in a cube: the face diagonal and the body diagonal.

### Step-by-Step Solution:

1. **Understanding the Cube Structure**:
- A cube has 12 edges, 6 faces, and 8 vertices.
- Each edge of the cube is of equal length \( a \).

2. **Calculating the Face Diagonal**:
- The face diagonal connects two opposite corners of a face of the cube.
- For a square face with side length \( a \), the length of the face diagonal can be calculated using the Pythagorean theorem:
\[ \text{Face Diagonal} = \sqrt{a^2 + a^2} = \sqrt{2a^2} = \sqrt{2}a \]

3. **Calculating the Body Diagonal**:
- The body diagonal connects two opposite corners of the cube that are not on the same face.
- To find the length of the body diagonal, we again use the Pythagorean theorem in three dimensions:
\[ \text{Body Diagonal} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = \sqrt{3}a \]

4. **Comparing the Diagonals**:
- We have calculated:
- Face Diagonal = \( \sqrt{2}a \)
- Body Diagonal = \( \sqrt{3}a \)
- Since \( \sqrt{3}a > \sqrt{2}a \), the body diagonal is longer than the face diagonal.

5. **Conclusion**:
- The largest diagonal in a cubic crystal of edge length \( a \) is the body diagonal, which measures \( \sqrt{3}a \).

### Final Answer:
The largest diagonal in a cubic crystal of edge length \( a \) is \( \sqrt{3}a \).

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