To tackle the task of drawing the unit cell for the face-centered cubic (FCC) lattice and identifying the specified crystallographic planes, let’s first understand the structure of the FCC lattice and the significance of crystallographic planes within it.

Understanding the FCC Lattice

The face-centered cubic (FCC) lattice is one of the most common crystal structures found in metals. In this arrangement, atoms are located at each of the corners of the cube and at the centers of each face. This results in a highly symmetrical structure that is known for its close packing and high atomic density.

Unit Cell Representation

To visualize the FCC unit cell, imagine a cube where:

• Atoms are positioned at each of the eight corners of the cube.
• Atoms are also located at the center of each of the six faces of the cube.

Each corner atom contributes 1/8th of an atom to the unit cell, and each face-centered atom contributes 1/2. Therefore, the total number of atoms per FCC unit cell is:

Number of atoms = 8 corners × 1/8 + 6 faces × 1/2 = 4 atoms

Identifying Crystallographic Planes

Crystallographic planes are defined by Miller indices, which are a notation system in crystallography for planes in crystal lattices. In an FCC lattice, common planes include:

• {100} - Planes that cut through the cube at the corners and face centers.
• {110} - Diagonal planes that intersect the edges of the cube.
• {111} - Planes that pass through the corners and face centers, representing the densest atomic packing.

Drawing the Unit Cell

When drawing the FCC unit cell, follow these steps:

1. Start with a cube outline.
2. Mark the positions of the corner atoms at each vertex of the cube.
3. Place an atom at the center of each face of the cube.
4. Label the corners as (0,0,0), (1,0,0), (0,1,0), etc., and the face-centered atoms accordingly.

Labeling the Planes

Once the unit cell is drawn, you can identify and label the crystallographic planes:

• For the {100} plane, draw a plane that cuts through the center of the cube, parallel to one of the faces.
• For the {110} plane, draw a diagonal line from one edge to the opposite edge, cutting through the cube.
• For the {111} plane, visualize a plane that intersects three corners of the cube and the center of the opposite face.

Final Thoughts

By accurately representing the FCC unit cell and the specified crystallographic planes, you gain a deeper understanding of the atomic arrangement and the properties that arise from these structures. This knowledge is fundamental in fields such as materials science and solid-state physics, where the arrangement of atoms significantly influences the material's characteristics.