The correct statement (s) for cubic close packed (ccp) three dimensional structure is (are)

Options

A. The number of the nearest neighbours of an atom present in the topmost layer is 12
B. The packing efficiency of atom is `74%`
C. The number of octahedral and tetrahedral voids per atom are 1 and 2, respectively
D. The unit cell edge length is `2sqrt(2)` times the radius of the atom

Answer

To determine the correct statements regarding the cubic close packed (ccp) three-dimensional structure, we will analyze each statement one by one.

### Step-by-Step Solution:

1. **Statement 1: The number of nearest neighbors of an atom present in the topmost layer is 12.**
- **Analysis**: In a cubic close-packed structure, each atom in the topmost layer has nearest neighbors in the layers directly above and below it. However, when considering only the topmost layer, the nearest neighbors are actually 6 (the atoms in the same layer) plus 6 from the layer below, totaling 12. However, the statement is misleading as it specifies "topmost layer" without considering the below layer. Thus, this statement is **false**.

2. **Statement 2: The packing efficiency of an atom is 74%.**
- **Analysis**: The packing efficiency for cubic close packing (ccp) or face-centered cubic (fcc) structures is indeed 74%. This is calculated based on the volume occupied by the atoms compared to the total volume of the unit cell. Thus, this statement is **true**.

3. **Statement 3: The number of octahedral and tetrahedral voids per atom are 1 and 2 respectively.**
- **Analysis**: In a face-centered cubic (fcc) structure, there is 1 octahedral void and 2 tetrahedral voids for every atom present. This is consistent with the known properties of the ccp structure. Therefore, this statement is **true**.

4. **Statement 4: The unit cell edge length is 2 times the radius of the atom.**
- **Analysis**: The relationship between the edge length (a) of the unit cell and the radius (r) of the atom in a ccp structure is given by the equation \(4r = \sqrt{2}a\). Rearranging this gives \(a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r\), which is not equal to 2r. Therefore, this statement is **false**.

### Summary of Correct Statements:
- The correct statements regarding the cubic close packed (ccp) structure are:
- Statement 2: The packing efficiency of an atom is 74%.
- Statement 3: The number of octahedral and tetrahedral voids per atom are 1 and 2 respectively.

### Final Answer:
The correct statements are 2 and 3.

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