Find the critical radius ratio for triangular void

Answer

To find the critical radius ratio for triangular voids, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Structure**:
Triangular voids are formed when three spheres are arranged in such a way that they create a triangular shape. The coordination number for this arrangement is 3.

2. **Defining Distances**:
- The distance between the centers of two adjacent spheres is denoted as \(2r\) (where \(r\) is the radius of the spheres).
- The distance from the center of one sphere to the center of the void formed is \(r + r = 2r\).

3. **Using Geometry**:
- In the triangular arrangement, we can visualize the triangle formed by the centers of the three spheres. The angle at the center of the void is \(30^\circ\).
- We can apply the cosine rule in this triangle.

4. **Applying the Cosine Function**:
- Using the cosine of the angle, we have:
\[ \cos(30^\circ) = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{r}{r + r} \]
- Here, the base is \(r\) and the hypotenuse is \(r + r = 2r\).

5. **Calculating the Cosine Value**:
- We know that \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\). Therefore, we can set up the equation:
\[ \frac{r}{2r} = \frac{\sqrt{3}}{2} \]

6. **Rearranging the Equation**:
- This simplifies to:
\[ r = \frac{\sqrt{3}}{2} \cdot 2r \]
- Which leads to:
\[ 1 = \sqrt{3} \]

7. **Finding the Critical Radius Ratio**:
- Rearranging gives us:
\[ r + r = 2r = \frac{2}{\sqrt{3}}R \]
- Thus, we can express the critical radius ratio as:
\[ \frac{r}{R} = \frac{1}{\sqrt{3}} \approx 0.577 \]

8. **Final Result**:
- The critical radius ratio for triangular voids is approximately \(0.577\).