Introduction to Unit Cells

Cubic Structures

• Simple Cubic Structure

  • 1 atom per unit cell
  • Coordination number: 6
  • Volume efficiency: 52% (48% empty)
  • Edge length (x) = 2r

• Body-Centered Cubic (BCC) Structure

  • 2 atoms per unit cell
  • Coordination number: 8
  • Volume efficiency: 68% (32% empty)
  • Edge length (x) = ( \frac{4}{\sqrt{3}} \times r )

• Face-Centered Cubic (FCC) Structure

  • 4 atoms per unit cell
  • Coordination number: 12
  • Volume efficiency: 74% (26% empty)
  • Edge length (x) = ( \sqrt{8} \times r )

Key Concepts

• Coordination Number: Number of atoms adjacent to a single atom.
• Edge Length (x): Side length of a cube.
• Atomic Radius (r): Half of the diameter of an atom.

Calculations

Simple Cubic Structure

• Atoms per Unit Cell: 8 corners with 1/8 of an atom each, totaling 1 atom.
• Coordination: Each atom is attached to 6 others (up, down, left, right, front, back).
• Volume Utilization: Derive 52% by comparing the atom's volume to the cube's volume.
  • ( \frac{\frac{4}{3} \pi r^3}{(2r)^3} = 0.5236 )

Body-Centered Cubic Structure

• Atoms per Unit Cell: 8 corners with 1/8 of an atom each plus 1 in the center, totaling 2 atoms.
• Coordination: Central atom connected to 8 corners.
• Volume Utilization: Calculate 68% efficiency.
  • ( \frac{2 \times \frac{4}{3} \pi r^3}{(\frac{4}{\sqrt{3}} r)^3} = 0.68 )
  • Steps: Simplify to ( \frac{2 \pi \sqrt{3}}{16} ) leading to ( 0.68 )

Additional Notes

• These calculations and concepts are crucial for understanding material properties and can be applied to calculate densities or atomic radii in specific structures.