In a close-packed structure, if r is the radius of the spherical void and R is the radius of the spheres forming voids, the critical r/R value for the tetrahedral void is equal to

Options

A. 0.155
B. 0.225
C. 0.414
D. 0.732

Answer

To solve the problem regarding the critical ratio of the radius of the spherical void (r) to the radius of the spheres forming the voids (R) in a close-packed structure, we will follow these steps:

### Step 1: Understand the Close-Packed Structure
In a close-packed structure, atoms are arranged in a way that maximizes the packing efficiency. There are two types of voids formed in such structures: tetrahedral and octahedral voids.

**Hint:** Remember that tetrahedral voids are formed when four spheres surround a central void, while octahedral voids are surrounded by six spheres.

### Step 2: Identify the Critical Ratios
For tetrahedral voids, the critical ratio of the radius of the void (r) to the radius of the spheres (R) is given by specific values. The critical ratio for tetrahedral voids is approximately between 0.225 and 0.414.

**Hint:** The ratio for tetrahedral voids is specifically defined, and you need to remember the range of values.

### Step 3: Determine the Valid Range
The critical value for the tetrahedral void is 0.225 < r/R < 0.414. This means that for a void to be classified as a tetrahedral void, the ratio must be greater than 0.225 but less than 0.414.

**Hint:** Focus on the boundaries of the ratio to determine the acceptable values for r/R.

### Step 4: Conclusion
Since the question asks for the critical r/R value for the tetrahedral void, we conclude that the critical ratio is 0.225. This means that the radius of the void (r) must be at least 0.225 times the radius of the spheres (R) to be considered a tetrahedral void.

**Hint:** The answer is based on the lower limit of the range for tetrahedral voids.

### Final Answer
The critical r/R value for the tetrahedral void is **0.225**.