Examples

Representing Miller and Miller–Bravais indices

This section demonstrates how to create various indices using the package.

julia> using MillerIndices
julia> Miller(1, 2, 3)  # Create Miller indices in real space3-element MillerIndices.Miller:
 1
 2
 3
julia> ReciprocalMiller(1, 2, 3)  # Create Miller indices in reciprocal space3-element MillerIndices.ReciprocalMiller:
 1
 2
 3
julia> MillerBravais(2, -1, -1, 3)  # Create Miller–Bravais indices in real space4-element MillerIndices.MillerBravais:
  2
 -1
 -1
  3
julia> ReciprocalMillerBravais(1, 0, -1, 0)  # Create Miller–Bravais indices in reciprocal space4-element MillerIndices.ReciprocalMillerBravais:
  1
  0
 -1
  0

Generating indices from strings

Utilize the m_str macro to rapidly generate indices from a string format.

julia> m"[-1, 0, 1]"  # Generate Miller indices from a string3-element MillerIndices.Miller:
 -1
  0
  1
julia> m"<2, -1, -1, 3>"  # Generate Miller–Bravais indices from a string4-element MillerIndices.MillerBravais:
  2
 -1
 -1
  3

Converting between Miller and Miller–Bravais indices

Switch between different index notations seamlessly.

julia> MillerBravais(Miller(-1, 0, 1))4-element MillerIndices.MillerBravais:
 -2
  1
  1
  3
julia> Miller(MillerBravais(-2, 1, 1, 3))3-element MillerIndices.Miller:
 -1
  0
  1

Finding equivalent directions/planes

Discover how to determine all directions/planes equivalent to a given index by symmetry.

julia> familyof(Miller(-1, 0, 1))6-element Vector{MillerIndices.Miller}:
 <-1 0 1>
 <0 -1 1>
 <1 1 1>
 <1 0 1>
 <0 1 1>
 <-1 -1 1>
julia> familyof(MillerBravais(-1, -1, 2, 3))6-element Vector{MillerIndices.MillerBravais}:
 <-1 -1 2 3>
 <-1 2 -1 3>
 <2 -1 -1 3>
 <1 1 -2 3>
 <1 -2 1 3>
 <-2 1 1 3>

Angle calculation

Determine the angle between two vectors using a metric tensor.

  1. Test the angle between $[1 2 1]$ and $[0 0 1]$ directions in gallium nitride:

    using CrystallographyBase, Unitful
a, c = 3.19u"angstrom", 5.19u"angstrom"
g = MetricTensor(a, a, c, 90, 90, 120)
θ = anglebtw(Miller(1, 0, 0), Miller(1, 1, 1), g)

  2. The length of the reciprocal lattice vector and the interplanar spacing:

    g = inv(MetricTensor(2, 4, 3, 90, 45, 90))
lengthof(ReciprocalMiller(1, 0, 2), g)
interplanar_spacing(ReciprocalMiller(1, 0, 2), g)

  3. Calculate the angle between two plane normals:

    g = inv(MetricTensor(4, 6, 5, 90, 135, 90))  # Monoclinic
𝐱 = ReciprocalMiller(1, 0, 1)
𝐲 = ReciprocalMiller(-2, 0, 1)
anglebtw(𝐱, 𝐲, g)