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, we can follow these steps:

### Step 1: Understand the Structure
In a cubic lattice, tetrahedral voids are formed by the arrangement of atoms at the corners and the face centers of the cube. Each tetrahedral void is surrounded by four atoms.

### Step 2: Identify the Edge Length
Let the edge length of the cube be denoted as \( a \).

### Step 3: Determine the Body Diagonal
The body diagonal of the cube can be calculated using the formula:
\[ \text{Body Diagonal} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3a^2} = a\sqrt{3} \]

### Step 4: Calculate the Distance Between Tetrahedral Voids
The tetrahedral voids are located at specific positions within the cube. To find the minimum distance between two tetrahedral voids, we consider the geometry of the tetrahedral voids.

1. The tetrahedral voids are located at the midpoints of the edges of the cube.
2. The distance between two tetrahedral voids can be visualized in terms of the body diagonal of the cube.

### Step 5: Use the Midpoint Concept
If we consider the body diagonal, we can find the distance between the tetrahedral voids by using the midpoint of the edges:
- The distance between two tetrahedral voids along the body diagonal is half of the body diagonal.

### Step 6: Final Calculation
From the body diagonal \( a\sqrt{3} \), the distance between the two tetrahedral voids can be calculated as:
\[ \text{Minimum Distance} = \frac{a\sqrt{3}}{2} \]

However, since we are looking for the minimum distance between two tetrahedral voids, we can simplify this further to:
\[ \text{Minimum Distance} = \frac{a}{2} \]

### Conclusion
Thus, the minimum distance between two tetrahedral voids in a cubic lattice with edge length \( a \) is:
\[ \text{Minimum Distance} = \frac{a}{2} \]