How to calculate coordination number and number of atoms per unit cell in types of crystals

Here's how to calculate the coordination number and the number of atoms per unit cell for common crystal types:

Coordination Number (CN)

The coordination number is the number of nearest neighbors (atoms directly touching) a central atom in the crystal lattice.

• Simple Cubic (SC): Each atom has 6 nearest neighbors.
  • CN = $\boxed{\text{6}}$
• Body-Centered Cubic (BCC): The central atom touches all 8 corner atoms.
  • CN = $\boxed{\text{8}}$
• Face-Centered Cubic (FCC) / Cubic Close-Packed (CCP): Each atom has 12 nearest neighbors.
  • CN = $\boxed{\text{12}}$
• Hexagonal Close-Packed (HCP): Each atom has 12 nearest neighbors.
  • CN = $\boxed{\text{12}}$

Number of Atoms Per Unit Cell (Z)

The number of atoms per unit cell is the effective total number of atoms contained within one unit cell, considering that atoms at corners, edges, or faces are shared with adjacent unit cells.

• Contribution of atoms based on their position:
  • Corner atom: Contributes $\frac{1}{8}$ to the unit cell.
  • Face-centered atom: Contributes $\frac{1}{2}$ to the unit cell.
  • Edge-centered atom: Contributes $\frac{1}{4}$ to the unit cell.
  • Body-centered atom: Contributes 1 (entirely within the unit cell).

Let's calculate Z for common cubic structures:

1. Simple Cubic (SC)

• Atoms at corners: 8 corners $\times \frac{1}{8}$ atom/corner = 1 atom
• Total atoms per unit cell: $$Z = 8 \times \frac{1}{8} = \boxed{\text{1 atom}}$$

2. Body-Centered Cubic (BCC)

• Atoms at corners: 8 corners $\times \frac{1}{8}$ atom/corner = 1 atom
• Atom at body center: 1 body-centered atom $\times 1$ atom/body = 1 atom
• Total atoms per unit cell: $$Z = \left(8 \times \frac{1}{8}\right) + (1 \times 1) = 1 + 1 = \boxed{\text{2 atoms}}$$

3. Face-Centered Cubic (FCC)

• Atoms at corners: 8 corners $\times \frac{1}{8}$ atom/corner = 1 atom
• Atoms at face centers: 6 faces $\times \frac{1}{2}$ atom/face = 3 atoms
• Total atoms per unit cell: $$Z = \left(8 \times \frac{1}{8}\right) + \left(6 \times \frac{1}{2}\right) = 1 + 3 = \boxed{\text{4 atoms}}$$

4. Hexagonal Close-Packed (HCP)
For a standard hexagonal unit cell (a prism containing three layers of atoms):

• Atoms at corners: 12 corners $\times \frac{1}{6}$ atom/corner = 2 atoms
• Atoms at top/bottom face centers: 2 faces $\times \frac{1}{2}$ atom/face = 1 atom
• Atoms entirely within the cell (middle layer): 3 atoms $\times 1$ atom/atom = 3 atoms
• Total atoms per unit cell: $$Z = \left(12 \times \frac{1}{6}\right) + \left(2 \times \frac{1}{2}\right) + (3 \times 1) = 2 + 1 + 3 = \boxed{\text{6 atoms}}$$