Question

Name and draw the three cubic unit cells. Describe their similarities and differences.

Short answer

The cubic unit cells are Simple Cubic, Body-Centered Cubic, and Face-Centered Cubic. They differ in atom arrangement and coordination numbers.

Step-by-step solution

Define the Simple Cubic Unit Cell

The Simple Cubic (SC) unit cell is the most basic cubic unit cell structure. It features one atom located at each of the eight corners of a cube. Each atom at a corner is shared among eight adjacent unit cells, resulting in effectively one atom per unit cell when accounting for the shared vertices. This structure has a coordination number of 6, which means each atom is in contact with six others.

Define the Body-Centered Cubic Unit Cell

The Body-Centered Cubic (BCC) unit cell has the same arrangement as the Simple Cubic structure but adds one additional atom in the center of the cube. This centering atom's position leads to a total of two atoms per unit cell: one in the center entirely within the unit cell and one contributed collectively from the corner atoms. The BCC has a higher coordination number of 8.

Define the Face-Centered Cubic Unit Cell

The Face-Centered Cubic (FCC) unit cell is more complex. Besides the corner atoms similar to the Simple Cubic and BCC structures, the FCC unit cell includes an additional atom on each of the six faces of the cube. These face-centered atoms are shared by two adjacent unit cells, giving a total of 4 atoms per unit cell. The FCC has the highest coordination number of 12.

Compare the Unit Cells

All three unit cells share a cubic geometry and have atoms located at the corners. The main differences lie in the placement of additional atoms, affecting the total number of atoms per unit cell and their coordination numbers. The Simple Cubic has only corner atoms, the BCC includes a central atom, and the FCC adds face-centered atoms.

Key concepts

Simple Cubic Unit Cell

A simple cubic unit cell is the most basic type of cubic structure in crystallography. Imagine a cube with an atom at each of its eight corners. These corner atoms are not entirely contained within a single unit cell. Instead, each corner atom is actually shared among eight adjacent unit cells. This sharing results in effectively only one atom per unit cell when you consider all shared portions.
This configuration leads to a coordination number of 6, meaning each atom in the structure touches six others. The simple cubic unit cell is rare in materials science because it is not the most efficient way to pack atoms. Yet, it provides a foundational understanding of more complex structures.

Body-Centered Cubic Unit Cell

The body-centered cubic (BCC) unit cell is like the simple cubic, but with an additional twist. Besides the eight corner atoms, there is one atom sitting right at the center of the cube. This central atom is unique because it belongs completely to that unit cell.
The presence of this central atom changes the dynamics of the structure. In total, a BCC unit cell has two atoms: one from the shared corner atoms and one whole central atom. The coordination number here is 8, as each atom in this structure is in contact with eight others. This arrangement allows for a more densely packed structure compared to the simple cubic configuration.

Face-Centered Cubic Unit Cell

The face-centered cubic (FCC) unit cell advances the idea of atomic placement even further. Similar to the simple cubic and BCC structures, it has atoms at each of the eight corners. Additionally, it includes an atom centered on each of the cube's six faces.
Because each face-centered atom is shared by two adjacent unit cells, this configuration results in a total of four atoms per unit cell. The coordination number, which is the number of nearest neighbors surrounding an atom, reaches its highest in the FCC structure at 12. This is due to the optimal packing efficiency that the face-centered arrangement offers, making it the most densely packed of the three cubic types.

Coordination Number

The concept of coordination number is crucial when discussing unit cells in crystal structures. It tells you how many neighbor atoms surround a given atom.
In a simple cubic unit cell, this number is 6. In a body-centered cubic unit cell, it is 8 due to the additional central atom providing more contact points. Lastly, in the face-centered cubic unit cell, the coordination number peaks at 12, maximizing atomic interaction.

• Simple Cubic: Coordination number of 6
• Body-Centered Cubic: Coordination number of 8
• Face-Centered Cubic: Coordination number of 12

Coordination numbers are essential for understanding the stability and properties of materials.

Crystal Structure

Crystal structures define how atoms are systematically arranged in a solid. In cubic unit cells, atoms follow a repeating pattern both within the unit cell and extending into larger three-dimensional structures.
The three principal types of cubic unit cells—simple cubic, body-centered cubic, and face-centered cubic—all exemplify different crystal structures based on how they pack their atoms. This arrangement affects the properties of the materials, including density, stability, and mechanical strength.
Understanding these structures is vital, as they form the basis of crystallography and material science, influencing everything from mineral formation to the creation of engineered materials.