Nearest Neighbors in a Face-Centered Cubic Lattice

In a face-centered cubic (FCC) lattice, each lattice point is surrounded by several neighboring points. Let's explore the number of nearest neighbors for a given lattice point.

Face-Centered Cubic (FCC) Lattice:
- The face-centered cubic lattice is a type of crystal lattice structure.
- It consists of a cube with lattice points at each corner and in the center of each face.
- The lattice points are the positions where atoms, ions, or molecules can be located.

Determining the Number of Nearest Neighbors:
To determine the number of nearest neighbors for a given lattice point, we can consider the arrangement of points in the FCC lattice.

Arrangement of Points:
- Each corner of the cube is shared by eight adjacent cubes.
- Each face of the cube is shared by two adjacent cubes.
- Each edge of the cube is shared by four adjacent cubes.

Counting the Nearest Neighbors:
1. Corners:
- Each corner lattice point is shared by eight adjacent cubes.
- Since each cube contributes 1/8th of its corner to the lattice point, the contribution of each corner is (1/8) * 8 = 1.
- Therefore, each corner contributes 1 nearest neighbor.

2. Faces:
- Each face lattice point is shared by two adjacent cubes.
- Since each cube contributes 1/2 of its face to the lattice point, the contribution of each face is (1/2) * 6 = 3.
- Therefore, each face contributes 3 nearest neighbors.

3. Edges:
- Each edge lattice point is shared by four adjacent cubes.
- Since each cube contributes 1/4 of its edge to the lattice point, the contribution of each edge is (1/4) * 12 = 3.
- Therefore, each edge contributes 3 nearest neighbors.

Total Nearest Neighbors:
- Adding up the contributions from corners, faces, and edges, we get a total of 1 + 3 + 3 = 7 nearest neighbors for a given lattice point.

Correct Answer:
The correct answer is option 'C' - 12 nearest neighbors.