A cubic unit cell is a fundamental building block in crystallography, characterized by its high symmetry and simple geometric parameters. It is defined by three equal edge lengths and three 90-degree angles between these edges, forming a perfect cube. Understanding its characteristics is key to comprehending the arrangement of atoms in many solid materials.
Defining Features of a Cubic Unit Cell
Cubic unit cells are among the most common and symmetrical types of crystal systems. Here are their primary defining features:
- Edge Lengths: All three crystallographic axes have equal lengths, denoted as a = b = c. This means the cube has sides of identical measurement.
- Interaxial Angles: All three angles between the crystallographic axes are 90 degrees, denoted as α = β = γ = 90°. This forms perpendicular edges.
- High Symmetry: Cubic systems possess a high degree of symmetry, including multiple axes of rotation and planes of reflection.
Types of Cubic Unit Cells
Within the cubic system, there are three main types of unit cells, distinguished by the positions of the atoms within the cube:
1. Primitive Cubic (PC) or Simple Cubic (SC)
The primitive cubic unit cell is the simplest arrangement.
- Atom Locations: In a primitive cubic unit cell, atoms are located only at the eight corners of the cube.
- Atom Contribution: Each atom at a corner is shared by eight adjacent unit cells (four in the same layer and four in the layers above/below). Therefore, only one-eighth (1/8) of an atom contributes to any single primitive cubic unit cell from each corner.
- Total Atoms per Unit Cell: (8 corners × 1/8 atom/corner) = 1 atom.
- Coordination Number: 6 (each atom is surrounded by 6 nearest neighbors).
- Packing Efficiency: Relatively low at 52%.
2. Body-Centered Cubic (BCC)
The body-centered cubic unit cell adds an atom to the center of the cube.
- Atom Locations: Atoms are present at all eight corners and one additional atom is located precisely at the center of the cube.
- Atom Contribution:
- Corner atoms: (8 corners × 1/8 atom/corner) = 1 atom.
- Body-centered atom: (1 atom × 1 atom/cell) = 1 atom.
- Total Atoms per Unit Cell: 1 (from corners) + 1 (body-centered) = 2 atoms.
- Coordination Number: 8 (each atom is surrounded by 8 nearest neighbors).
- Packing Efficiency: Higher than primitive cubic, at 68%. Examples include alkali metals like sodium and potassium, and transition metals like iron at room temperature. Learn more about BCC structures at LibreTexts Chemistry.
3. Face-Centered Cubic (FCC)
The face-centered cubic unit cell has atoms on each face.
- Atom Locations: Atoms are located at all eight corners and at the center of each of the six faces of the cube.
- Atom Contribution:
- Corner atoms: (8 corners × 1/8 atom/corner) = 1 atom.
- Face-centered atoms: Each atom on a face is shared by two unit cells, contributing one-half (1/2) to each cell. (6 faces × 1/2 atom/face) = 3 atoms.
- Total Atoms per Unit Cell: 1 (from corners) + 3 (from faces) = 4 atoms.
- Coordination Number: 12 (each atom is surrounded by 12 nearest neighbors).
- Packing Efficiency: The highest among the cubic systems, at 74%, making it a highly efficient packing arrangement. Many common metals like copper, aluminum, silver, and gold crystallize in an FCC structure. For a detailed look, refer to Wikipedia on Face-centered cubic.
Summary Table of Cubic Unit Cell Characteristics
| Characteristic | Primitive Cubic (PC/SC) | Body-Centered Cubic (BCC) | Face-Centered Cubic (FCC) |
|---|---|---|---|
| Atom Locations | Corners only | Corners + 1 in body center | Corners + 1 in each face center |
| Atoms per Unit Cell | 1 | 2 | 4 |
| Coordination Number | 6 | 8 | 12 |
| Packing Efficiency | 52% | 68% | 74% |
| Edge Length (a) | 2r (where r is atomic radius) | 4r/√3 | 2√2r |
These distinct characteristics make cubic unit cells fundamental in describing the crystal structures of a vast range of materials, influencing their physical and chemical properties.
