Minimum distance between two tetrahedral voids if a is the edge length of the cube is

Options

A. `a/4`
B. `a/(2sqrt2)`
C. `a/2`
D. `sqrt(3a)/4`

Answer

To find the minimum distance between two tetrahedral voids in a cubic lattice where \( a \) is the edge length of the cube, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Structure**: In a cubic lattice, tetrahedral voids are formed at specific positions relative to the atoms in the lattice. Each tetrahedral void is surrounded by four atoms.

2. **Identify Tetrahedral Voids**: In a face-centered cubic (FCC) structure, tetrahedral voids are located at the following coordinates:
- \( (0, 0, 0) \)
- \( \left( \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \right) \)
- \( \left( \frac{3}{4}, \frac{3}{4}, \frac{3}{4} \right) \)
- Other equivalent positions can also be considered.

3. **Calculate the Distance**: To find the minimum distance between two tetrahedral voids, we can consider the void at \( (0, 0, 0) \) and the void at \( \left( \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \right) \).

The distance \( d \) between these two points can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{\left(\frac{1}{4} - 0\right)^2 + \left(\frac{1}{4} - 0\right)^2 + \left(\frac{1}{4} - 0\right)^2} \]
\[ = \sqrt{\left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^2 + \left(\frac{1}{4}\right)^2} \]
\[ = \sqrt{3 \cdot \left(\frac{1}{4}\right)^2} \]
\[ = \sqrt{3} \cdot \frac{1}{4} \]
\[ = \frac{\sqrt{3}}{4} \]

4. **Relate to Edge Length**: Since the edge length \( a \) of the cube relates to the positions of the voids, we can express the distance in terms of \( a \). The minimum distance between two tetrahedral voids is given as:
\[ \text{Minimum distance} = \frac{a}{2} \]

5. **Conclusion**: Therefore, the minimum distance between two tetrahedral voids in a cubic lattice with edge length \( a \) is:
\[ \frac{a}{2} \]

### Final Answer:
The minimum distance between two tetrahedral voids is \( \frac{a}{2} \).