The total number of octahedral void (s) per atom present in a cubic close packed structure is

Options

A. 2
B. 4
C. 1
D. 3

Answer

To find the total number of octahedral voids per atom present in a cubic close packed (CCP) structure, we can follow these steps:

### Step 1: Understand the structure of CCP
Cubic close packed (CCP) structure is also known as face-centered cubic (FCC) structure. In this arrangement, atoms are located at each corner and the centers of all the faces of the cube.

### Step 2: Identify the locations of octahedral voids
In a CCP structure, octahedral voids are located:
- At the body center of the cube
- At the edge centers of the cube

### Step 3: Count the octahedral voids
1. **Body Center**: There is 1 octahedral void at the body center of the cube.
2. **Edge Centers**: There are 12 edges in the cube, and each edge contributes 1 octahedral void. However, each edge center is shared by 4 adjacent cubes, so the contribution from each edge center is \( \frac{1}{4} \).

Therefore, the total contribution from the edge centers is:
\[ 12 \text{ edges} \times \frac{1}{4} = 3 \]

### Step 4: Calculate total octahedral voids
Now, we can add the contributions from the body center and the edge centers:
\[ \text{Total octahedral voids} = 1 \text{ (from body center)} + 3 \text{ (from edge centers)} = 4 \]

### Step 5: Determine the number of atoms in CCP
In a cubic close packed structure, there are a total of 4 atoms per unit cell. This is because:
- Each corner atom contributes \( \frac{1}{8} \) (8 corners)
- Each face-centered atom contributes \( \frac{1}{2} \) (6 faces)

Calculating this gives:
\[ \text{Total atoms} = 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 1 + 3 = 4 \]

### Step 6: Calculate octahedral voids per atom
To find the number of octahedral voids per atom, we divide the total number of octahedral voids by the total number of atoms:
\[ \text{Octahedral voids per atom} = \frac{\text{Total octahedral voids}}{\text{Total atoms}} = \frac{4}{4} = 1 \]

### Final Answer
The total number of octahedral voids per atom present in a cubic close packed structure is **1**.

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