The number of octahedral sites per sphere in a fcc structure is :

Options

A. 8
B. 4
C. 2
D. 1

Answer

To determine the number of octahedral sites per sphere in a face-centered cubic (FCC) structure, we can follow these steps:

### Step 1: Understand the FCC Structure
In a face-centered cubic (FCC) structure, the atoms are located at:
- All eight corners of the cube
- The centers of all six faces of the cube

### Step 2: Count the Atoms in the FCC Unit Cell
- Each corner atom contributes \( \frac{1}{8} \) of an atom to the unit cell (since each corner is shared by 8 unit cells).
- There are 8 corners, so the contribution from the corners is:
\[ 8 \times \frac{1}{8} = 1 \text{ atom} \]
- Each face-centered atom contributes \( \frac{1}{2} \) of an atom to the unit cell (since each face is shared by 2 unit cells).
- There are 6 faces, so the contribution from the face-centered atoms is:
\[ 6 \times \frac{1}{2} = 3 \text{ atoms} \]
- Therefore, the total number of atoms in the FCC unit cell is:
\[ 1 + 3 = 4 \text{ atoms} \]

### Step 3: Identify the Octahedral Sites
In an FCC structure, octahedral voids are located:
- At the center of the cube (1 site)
- At the edge centers (12 edges, each contributing \( \frac{1}{4} \) of a site)

### Step 4: Calculate the Contribution from Edge Centers
- The contribution from the edge centers is:
\[ 12 \times \frac{1}{4} = 3 \text{ sites} \]

### Step 5: Total Number of Octahedral Sites
- The total number of octahedral sites in an FCC unit cell is:
\[ 1 \text{ (from the body center)} + 3 \text{ (from the edge centers)} = 4 \text{ octahedral sites} \]

### Step 6: Calculate the Number of Octahedral Sites per Sphere
- Since there are 4 atoms in the FCC structure and 4 octahedral sites, the number of octahedral sites per atom (or sphere) is:
\[ \frac{4 \text{ octahedral sites}}{4 \text{ atoms}} = 1 \text{ octahedral site per atom} \]

### Final Answer
The number of octahedral sites per sphere in an FCC structure is **1**.

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