The number of atoms per unit cell is calculated by considering the contribution of each atom based on its location within the unit cell. In a cubic crystal structure, atoms located at different positions contribute differently to a single unit cell.
Understanding Atomic Contributions
Different positions within the unit cell contribute differently to the overall count of atoms in that cell:
- Corner Atoms: Each corner atom is shared among eight adjacent unit cells. Therefore, each corner atom contributes only 1/8th of an atom to a single unit cell.
- Face-Centered Atoms: Atoms located at the center of each face are shared by two unit cells. They each contribute 1/2 of an atom to a single unit cell.
- Edge Atoms: Atoms located on an edge are shared by four unit cells. They each contribute 1/4 of an atom to a single unit cell.
- Body-Centered Atoms: An atom located at the very center of the unit cell is not shared and contributes a full 1 atom.
Calculating Atoms per Unit Cell in a Cubic Structure
The reference provided states that there are 8 atoms at the corners of each cubic unit cell, each contributing 1/8th to the cell. Thus the total number of atoms from the corners in a cubic cell is 8 x 1/8 = 1 atom. We can expand on this:
| Atom Location | Number of Atoms | Contribution per Atom | Total Atoms from Location |
|---|---|---|---|
| Corners | 8 | 1/8 | 1 |
| Face Centers | 6 | 1/2 | 3 |
| Edge Centers | 12 | 1/4 | 3 |
| Body Center | 1 | 1 | 1 |
Example: Simple Cubic (SC) Structure
A simple cubic structure has only corner atoms. Thus:
- Total atoms = (8 corners) * (1/8 contribution per corner) = 1 atom per unit cell.
Example: Face-Centered Cubic (FCC) Structure
A face-centered cubic structure has corner atoms and face-centered atoms. Thus:
- Total atoms = (8 corners 1/8) + (6 face centers 1/2) = 1 + 3 = 4 atoms per unit cell.
Example: Body-Centered Cubic (BCC) Structure
A body-centered cubic structure has corner atoms and one body-centered atom. Thus:
- Total atoms = (8 corners 1/8) + (1 body center 1) = 1 + 1 = 2 atoms per unit cell.
Conclusion
To calculate the total number of atoms per unit cell, you need to sum the contributions from all atom locations in the structure, based on their individual fractional contribution to the cell. This calculation is important for understanding the physical properties of a material.
