Lattice Parameters to Know for Mathematical Crystallography

Lattice parameters are key to understanding crystal structures in Mathematical Crystallography. They include unit cell dimensions, interaxial angles, and symmetry, which together define the size, shape, and arrangement of crystals, influencing their properties and behaviors.

1. Unit cell dimensions (a, b, c)

   • Represent the lengths of the edges of the unit cell in three-dimensional space.
   • Essential for defining the size and shape of the crystal lattice.
   • The parameters can vary significantly between different crystal structures.

2. Interaxial angles (α, β, γ)

   • Define the angles between the edges of the unit cell.
   • Critical for determining the geometry of the unit cell.
   • Combined with unit cell dimensions, they fully describe the unit cell's orientation in space.

3. Crystal systems

   • Classify crystals based on their unit cell dimensions and interaxial angles.
   • There are seven distinct crystal systems: cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, and triclinic.
   • Each system has unique symmetry properties and lattice types.

4. Bravais lattices

   • There are 14 distinct Bravais lattices that represent all possible lattice arrangements in three-dimensional space.
   • Each lattice is defined by its unit cell dimensions and symmetry.
   • Understanding Bravais lattices is crucial for identifying crystal structures.

5. Miller indices

   • A notation system to describe the orientation of lattice planes in a crystal.
   • Indices are derived from the reciprocal of the intercepts that the plane makes with the axes.
   • Essential for understanding crystallographic directions and planes.

6. Symmetry operations

   • Include translations, rotations, reflections, and inversions that can be applied to a crystal structure.
   • Help in determining the symmetry of the crystal and its properties.
   • Symmetry operations are fundamental in classifying crystals and understanding their behavior.

7. Space groups

   • Combinations of symmetry operations that describe the symmetry of a crystal structure.
   • There are 230 unique space groups in three dimensions.
   • Space groups are crucial for predicting physical properties and behaviors of crystals.

8. Reciprocal lattice

   • A mathematical construct used to describe the periodicity of a crystal in momentum space.
   • Helps in understanding diffraction patterns and electronic properties of materials.
   • The reciprocal lattice is essential for interpreting X-ray diffraction data.

9. Lattice planes and d-spacing

   • Lattice planes are imaginary planes that pass through lattice points in a crystal.
   • The distance between these planes is known as d-spacing, which is critical for diffraction studies.
   • d-spacing is inversely related to the Miller indices and is used to calculate angles in diffraction experiments.

10. Primitive and non-primitive unit cells

    • Primitive unit cells contain only one lattice point and are the smallest repeating unit.
    • Non-primitive unit cells contain multiple lattice points and can simplify the description of complex structures.
    • Understanding the distinction is important for crystallography and material science applications.