Introduction to Crystallography

Crystallography

• Crystallography is the study of crystals and crystallized materials found in nature.
• These materials can exist in single forms, or groups of forms.
• Crystallography focuses on the shape of crystals, the relationship between crystal faces, and various natural properties.
• A crystal is a part of a solid medium that has a homogeneous chemical composition.
• It's defined by flat surfaces with symmetrical relationships.
• Crystals are formed by natural forces under specific pressure and temperature conditions.
• Crystals possess a specific atomic structure.

Crystal Characteristics

• Crystal faces: These are the outer surfaces of a crystal. They shape the crystal, creating a regular geometry, a result of the crystal's internal atomic arrangement. Typically, crystal faces are flat.
• Edges: These are lines formed where crystal planes or faces meet.
• Interfacial angles: These angles are created by the convergence of two crystalline faces.
• Solid angles: These angles result from the convergence of more than two crystal faces.

Crystal Symmetry

• Crystal symmetry refers to the process that results in certain crystal facets occupying the same positions, as if by repetition.
• Symmetry operations are the repetitive operations applied to produce symmetry.
• Elements of symmetry include:
  • Planes of symmetry: These planes divide the crystal into two symmetrical halves. Each face, edge, or corner on one side of the plane has a similar counterpart on the other side.
  • Axes of symmetry:
    • Rotation axis of symmetry: An imaginary line through the crystal center, rotating the crystal. A face, letter or corner angle is repeated upon rotation by specific incremental degrees:
      • Diagonal axis: rotates 180 degrees.
      • Trigonal axis: rotates 120 degrees
      • Tetragonal axis rotates 90 degrees
      • Hexagonal axis: rotates 60 degrees.
    • Rotation-inversion axis of symmetry: Involves a rotation followed by an inversion (flipping) through a center point.
  • Crystal symmetry center: An imaginary central point around which crystal faces, etc. are arranged in pairs, with similar positions on opposite sides, equidistant.

Crystal Systems

• Crystal systems are groups of crystals that share similar shapes, areas, and physical/chemical properties.
• They are categorized based on their axial systems.
• There are seven crystal systems.
  • Cubic
  • Tetragonal
  • Orthorhombic
  • Monoclinic
  • Triclinic
  • Trigonal (Rhombohedral)
  • Hexagonal
• These systems have specific relationships between the axial lengths (a, b, c) and the angles (α, β, γ) between them.

C-Axis

• The c-axis of a crystal can indicate the crystal system:
  • Hexagonal symmetry if the c-axis is hexagonal.
  • Trigonal symmetry if the c-axis is trigonal.
  • Quadrilateral symmetry if the c-axis is quadrilateral.
  • Binary symmetry if the c-axis is binary, which indicates an orthorhombic system.

Stereographic Projection

• Stereographic projection is a method to represent three-dimensional crystal data on a two-dimensional plane (a flat surface).
• Crystals are placed at the center of a sphere, and the normals to crystal faces intersect the sphere, defining points ("poles").
• The angles between poles on the sphere correlate to angles between faces in the crystal.