In hexagonal close packing of spherer in three dimensions.

Options

A. In one unit cell there are `12` octahedral voids and all are completely inside the unit cell.
B. In one unit cell there are six octahedral voids and all are competely inside the unit cell.
C. In one unit cell there are six octahedral voids and of which three are completely inside the unit cell and other three are from contributions of octahedral voids which are partially inside the unit cell
D. in one unit cell there are `12` tetrahedral voids, all are completely inside the unit cell.

Explanation

To solve the question regarding hexagonal close packing (HCP) of spheres in three dimensions, we will analyze the properties of the unit cell and the voids present in it.

### Step-by-Step Solution:

1. **Understanding Hexagonal Close Packing (HCP)**:
- HCP is a type of crystal structure where spheres are packed closely together. In this arrangement, each sphere touches 12 other spheres.

2. **Identifying the Number of Voids**:
- In HCP, there are two types of voids: octahedral voids and tetrahedral voids.
- The relationship between the number of octahedral voids (X) and tetrahedral voids is given by:
- Number of tetrahedral voids = 2 * Number of octahedral voids.

3. **Calculating the Number of Voids**:
- For HCP, the number of octahedral voids is always 6.
- Therefore, the number of tetrahedral voids will be:
\[ \text{Number of tetrahedral voids} = 2 \times 6 = 12. \]

4. **Analyzing Each Statement**:
- **Statement 1**: "In one unit cell there are 12 octahedral voids and are completely inside the unit cell."
- This statement is **false** because there are only 6 octahedral voids.
- **Statement 2**: "In one unit cell there are six octahedral voids and are completely inside the unit cell."
- This statement is **true** as it correctly states the number of octahedral voids.
- **Statement 3**: "In one unit cell there are six octahedral voids and of which three are completely inside the unit cell and other three are from contributions of octahedral voids which are partially inside the unit cell."
- This statement is **false** because all octahedral voids are completely inside the unit cell.
- **Statement 4**: "In one unit cell there are twelve tetrahedral voids, all are completely inside the unit cell."
- This statement is **true** as it correctly states the number of tetrahedral voids.

5. **Conclusion**:
- The correct statements are:
- Statement 2: True
- Statement 4: True

### Final Answer:
The true statements regarding hexagonal close packing of spheres in three dimensions are the second and fourth statements.

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