Miller Indices and Lattice Directions
A crystallographic direction is specified by a set of integers [h k l] — the Miller indices — which give the components of the direction vector in the crystal frame, expressed as multiples of the lattice translation vectors a, b, c. In a cubic crystal the direction [hkl] is simply the Cartesian vector (h, k, l), normalised to lie on the unit sphere:
d̂ = (h a + k b + l c) / |h a + k b + l c|
In non-cubic systems the metric tensor must be used to convert between lattice indices and Cartesian coordinates, since the lattice vectors are not orthonormal. The direction [hkl] is then the normalised version of the contravariant vector transformed by the metric tensor.
Angular Relationships Between Directions
The angle between two crystallographic directions [h₁k₁l₁] and [h₂k₂l₂] is found from the dot product of their unit vectors. In a cubic lattice this simplifies to:
cos φ = (h₁h₂ + k₁k₂ + l₁l₂) / (√(h₁²+k₁²+l₁²) · √(h₂²+k₂²+l₂²))
Symmetry-equivalent directions — those related by point-group operations — all subtend the same angles to any given direction and appear at the same set of positions on the unit sphere. Families of equivalent directions are denoted with angle brackets, e.g. ⟨110⟩ for the twelve face-diagonal directions of a cubic crystal.
In the Simulation
The viewer plots crystallographic directions as points on a 3D unit sphere and allows picking and comparing any two directions interactively:
Miller index input — type [hkl] directly; the corresponding point appears on the sphere ·
Second direction d₂ — add a second direction and read out the inter-direction angle φ ·
Symmetry equivalent directions — display the full family ⟨hkl⟩ of equivalent directions as additional points ·
Crystal system selector — observe how the spacing of directions changes with crystal symmetry.
Understanding directions on the sphere is the prerequisite for stereographic projection and pole figures, covered in the next submodule.