The distance between any two `TV_(s)` formed one any body diagonal of a closest packed structure is `= x xx a`, where `a` is the edge length of closest packed structure.
the value of `x = ……… .

Answer

To solve the problem, we need to find the value of \( x \) in the equation that relates the distance between two tetrahedral voids along a body diagonal of a closest packed structure to the edge length \( a \).

### Step-by-Step Solution:

1. **Understanding the Structure**:
- In a closest packed structure, tetrahedral voids are located at specific positions relative to the corners of the unit cell.

2. **Identify the Body Diagonal**:
- The body diagonal of a cube with edge length \( a \) can be calculated using the formula:
\[ \text{Length of body diagonal} = \sqrt{a^2 + a^2 + a^2} = \sqrt{3}a \]

3. **Position of Tetrahedral Voids**:
- The tetrahedral voids are positioned along the body diagonal. Specifically, each tetrahedral void is located at a distance of \( \frac{\sqrt{3}}{4}a \) from each corner along the body diagonal.

4. **Calculating the Distance Between Tetrahedral Voids**:
- Since there are two tetrahedral voids along the body diagonal, we need to calculate the total distance between them.
- The distance from one corner to the first tetrahedral void is \( \frac{\sqrt{3}}{4}a \).
- The distance from the other corner to the second tetrahedral void is also \( \frac{\sqrt{3}}{4}a \).
- Therefore, the total distance between the two tetrahedral voids is:
\[ \text{Total distance} = \sqrt{3}a - 2 \left(\frac{\sqrt{3}}{4}a\right) \]

5. **Simplifying the Expression**:
- Simplifying the above expression:
\[ \text{Total distance} = \sqrt{3}a - \frac{\sqrt{3}}{2}a = \left(\sqrt{3} - \frac{\sqrt{3}}{2}\right)a = \left(\frac{2\sqrt{3}}{2} - \frac{\sqrt{3}}{2}\right)a = \frac{\sqrt{3}}{2}a \]

6. **Relating to the Given Equation**:
- From the problem, we have the distance between the two tetrahedral voids as \( x \cdot a \).
- Setting the two expressions equal gives:
\[ x \cdot a = \frac{\sqrt{3}}{2}a \]

7. **Solving for \( x \)**:
- Dividing both sides by \( a \) (assuming \( a \neq 0 \)):
\[ x = \frac{\sqrt{3}}{2} \]

8. **Calculating the Numerical Value**:
- The numerical value of \( \frac{\sqrt{3}}{2} \) is approximately \( 0.866 \).

### Final Answer:
The value of \( x \) is \( 0.866 \).

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