Classification of Lattices

Introduction

• Crystal: Periodic arrangement of atoms.
• Lattice: Periodic set of points.
• Importance of classifying lattices into well-defined systems.

Classification Systems

• Seven Crystal Systems: Method to classify crystals.
  • Cubic
  • Tetragonal
  • Orthorhombic
  • Hexagonal
  • Trigonal (Rhombohedral)
  • Monoclinic
  • Triclinic
• Fourteen Bravais Lattices: Another classification method.

Seven Crystal Systems

• Cubic: All sides (a, b, c) and angles (α, β, γ) are equal. Units cells are cubes.
• Tetragonal: Angles are 90°, two sides are equal.
• Orthorhombic: Angles are 90°, but no sides are equal.
• Configurations for other systems are detailed.

Fourteen Bravais Lattices

• Primitive (P): Lattice points only at corners. Present in all 7 systems.
• Body-Centered (I): Points at corners and center. Found in cubic, tetragonal, orthorhombic.
• Face-Centered (F): Points at corners and face centers. Found in cubic and orthorhombic.
• End/ Base-Centered (C): Corners and one pair of parallel faces. Found in orthorhombic and monoclinic.
• Summary: 7 primitive + 3 body-centered + 2 face-centered + 2 base-centered = 14 Bravais Lattices.

Visualizing Lattices in the Cubic System

• Simple Cubic (Cubic P): Points at corners.
• Body-Centered Cubic (Cubic I): Points at corners and center.
• Face-Centered Cubic (Cubic F): Points at corners and centers of all faces.

Base-Centered Example in Orthorhombic

• Orthorhombic C: Lattice points at corners and one pair of parallel faces.
  • Analysis of axes: A long x, B long y, C long z.

Lack of Edge-Centered Lattices

• Edge-Centered: Not a valid lattice type.
  • Example: When considering edge-centered in a cube, points are not translationally equivalent (e.g., points A and B).

List of Bravais Lattices

• Cubic: P, I, F.
• Tetragonal: P, I.
• Orthorhombic: P, I, F, C.
• Hexagonal: P.
• Trigonal: P.
• Monoclinic: P, C.
• Triclinic: P.

Unanswered Questions

• Why are some lattices missing (e.g., Cubic C, Tetragonal F)?
• Empty boxes in the potential lattice table to be explored in the next video.