Introduction to Crystallography

Crystallography

• Crystallography is the study of crystals and crystallized materials found in nature.
• It examines the shape of crystals, the relationships between crystal faces, and various natural properties.
• Crystals are part of a solid medium with a homogeneous chemical composition.
• They are defined by smooth flat surfaces with symmetrical relations.
• Crystals form under specific pressure and temperature conditions and possess a certain atomic structure.

Crystal Characteristics

• Crystal faces: The outer surfaces of a crystal, determining its shape and reflecting the internal atomic arrangement. Typically flat.
• Edges: Lines formed by the meeting of two adjacent crystal faces.
• Interfacial angles: The angles formed by the convergence of two adjacent faces.
• Solid angles: Angles formed by convergence of more than two faces.

Crystal Symmetry

• Crystal symmetry describes how crystal facets repeat in a specific crystal group.
• All repetitive operations are classified as symmetry operations; symmetry involves repeating a pattern through specific rules.
• Elements of symmetry:
  • Planes of symmetry: A plane dividing the crystal into two identical halves (mirrored).
  • Axis of symmetry: An imaginary line through the crystal, rotating the crystal to repeat the same pattern. These include:
    • Rotational axis of symmetry: Rotating the crystal, a face, letter or corner repeats itself twice or more in a 360-degree rotation without displacement.
      • 2-Fold: 180-degree rotation
      • 3-Fold: 120-degree rotation
      • 4-Fold: 90-degree rotation
      • 6-Fold: 60-degree rotation
    • Rotation inversion symmetry axis: A combination of rotation & inversion through a center gives a symmetric shape, this means that the facets, letters or angular positions of the crystals can return to the same position.
  • Crystal symmetry center: An imaginary center within the crystal; related crystal faces, letters, and angles are in opposite positions and equal distances opposite the center.

Crystal Systems

• Crystal systems: Groups of similar crystals based on their axial and angular relationships.

  • Crystallographic axes: Imaginary lines intersecting at the crystal center, representing the axes of a crystal. These define shape.
  • Axial angles: Angles between the crystallographic axes in different crystal systems. These angles measure the relationships between the axes.
    • 3-axis Systems: Cubic, tetragonal, orthorhombic, monoclinic, triclinic have these values.
    • 4-axis Systems: Hexagonal and trigonal also have these.

• Crystal Form: A group of similar faces in a crystal (shape, area, physical, and chemical properties).

• Important note: Different crystal structures, lattices, and morphologies based on a unified system of axes belong to the same crystal system.

C-axis and Crystal Symmetry

• The c-axis can reveal information about the symmetry of the crystal.
• A crystal with a hexagonal c-axis has hexagonal symmetry.
• Other c-axis orientations may indicate trigonal, quadrilateral, binary symmetries, or no symmetry. The presence of certain symmetries (binary, etc.) and characteristics of the b axis can provide clues to classify the system (monoclinic, etc).

Symmetry Formula and Stereographic Projections

• Symmetry elements can be expressed using a symmetry formula.
• Stereographic projections are used to represent three-dimensional crystal structures on a 2D plane. They illustrate the relationships between crystal faces through representing the normal to faces as points on a sphere.

Crystal System Summaries (Examples from text)

• Cubic (a=b=c; α=β=γ=90°)
• Tetragonal (a=b≠c; α=β=γ=90°)
• Orthorhombic (a≠b≠c; α=β=γ=90°)
• Monoclinic (a≠b≠c; α=γ=90°; β≠90°)
• Triclinic (a≠b≠c; α≠β≠γ≠90°)
• Trigonal (a1=a2=a3≠c; α=β=90°; γ=120°)
• Hexagonal (a1=a2=a3≠c; α=β=90°; γ=120°)