L3 - Lattices and Symmetry

Lattices and Symmetry

Describing the positions of all atoms in a crystal is complex due to their vast number. However, crystal structures repeat periodically, which makes this task achievable.

Lattices

The repeating pattern of atomic arrangements in a mineral can be represented as a series of points called a lattice. Lattice points are connected via lattice vectors. A lattice can be constructed by:

  1. Choosing an origin.

  2. Specifying the lattice vectors.

    • Lattice vectors join any two lattice points

    • Unit cell vectors are the sides of the unit cell, and any lattice vector can be represented as a combination of unit cell vectors

  3. Placing a point at the end of each vector.

Unit Cells

A lattice can be broken down into a basic repeating shape called a unit cell. Unit cells placed next to each other reconstruct the lattice.

  • Primitive Unit Cell: Has lattice points only at each corner.

  • Non-Primitive Unit Cell: Has lattice points at the corners and elsewhere, providing a more descriptive representation of the crystal's symmetry and properties.

2D Bravais Lattices

In two dimensions, there are five fundamental types of lattices, known as Bravais lattices:

  • Square: |a| = |b|, γ = 90˚

  • Rectangular: |a| ≠ |b|, γ = 90˚

  • Oblique: |a| ≠ |b|, γ ≠ 90˚

  • Hexagonal: |a| = |b|, γ = 120˚

  • Centered Rectangular: |a| ≠ |b|, γ = 90˚, centered

3D Bravais Lattices

In three dimensions, there are fourteen fundamental types of lattices. This is more complex than in 2D because atoms can exist in the center of the volume of a unit cell (body-centered).

  • Cubic: |a| = |b| = |c|, α = β = γ = 90˚, primitive, body-centered, face-centered

  • Tetragonal: |a| = |b| ≠ |c|, α = β = γ = 90˚, primitive, body-centered

  • Orthorhombic: |a| ≠ |b| ≠ |c|, α = β = γ = 90˚, primitive, single-face centered, body-centered, face-centered

  • Monoclinic: |a| ≠ |b| ≠ |c|, α = γ = 90˚, β ≠ 90˚, primitive, body-centered

  • Hexagonal/trigonal: |a| = |b| ≠ |c|, α = β = 90 ˚, γ = 120˚

  • Triclinic: |a| ≠ |b| ≠ |c|, α ≠ β ≠ γ ≠ 90˚

Convolution

The lattice and unit cell describe the underlying symmetry, but don't specify the atoms present. To include atomic information, a motif of atoms (the arrangement of atoms) is placed on each lattice point. Mathematically, this 'copy-and-paste' function is called convolution. Convolving the arrangement of atoms with the lattice results in a crystal structure.

Symmetry Operations

Different crystal systems exhibit different symmetry operations:

  • Translational Symmetry: Moving an atom to an identical atom in an identical environment.

  • Rotational Symmetry: Rotating about an axis moves an atom to an equivalent atom in a rotated environment. Permitted rotations include 2-fold (diad), 3-fold (triad), 4-fold (tetrad), and 6-fold (hexad) rotations.

  • Mirror Symmetry: Reflecting an atom in a plane produces the reflected environment.

Each Bravais lattice possesses a characteristic symmetry described by rotations.

Miller Indices

Many mineral properties manifest on 2D planes within the 3D crystal. These planes are described using Miller indices, which are a series of three numbers indicating the denominators of the intersections of the plane with each axis (x, y, and z). These planes are crucial for:

  • X-ray diffraction (used to determine crystal structures).

  • Crystal shape and cleavage.