Groups

All groups in Crystalline are concrete instances of the abstract supertype Crystalline.AbstractGroup{D}, referring to a group in D dimensions. Crystalline.AbstractGroup{D} is itself a subtype of AbstractVector{SymOperation{D}}. Crystalline currently supports five group types: SpaceGroup, PointGroup, LittleGroup, SubperiodicGroup, SiteGroup, and MSpaceGroup.

Example: space groups

The one, two, and three-dimensional space groups are accessible via spacegroup, which takes the space group number sgnum and dimension D or Val(D) as input and returns a SpaceGroup{D}:

using Crystalline

D     = 3  # dimension
sgnum = 16 # space group number (≤2 in 1D, ≤17 in 2D, ≤230 in 3D)
sg    = spacegroup(sgnum, D) # or `spacegroup(sgnum, Val(D))`

SpaceGroup{3} ⋕16 (P222) with 4 operations:
 1
 2₀₀₁
 2₀₁₀
 2₁₀₀

Where practical, spacegroup should be called with a Val(D) dimension to ensure type stability; here and elsewhere in the documentation, we will often use D::Int instead for simplicity.

By default, the returned operations are given in the conventional setting of the International Tables of Crystallography, Volume A (ITA). Conversion to a primitive basis (in the CDML setting) can be accomplished via primitivize.

In addition to space groups, Crystalline.jl provides access to the operations of point groups (pointgroup), little groups (littlegroups), subperiodic groups (subperiodicgroup; including rod, layer, and frieze groups), site symmetry groups (sitegroup and sitegroups), and magnetic space groups (mspacegroup).

Multiplication tables

We can compute the multiplication table of a space group (under the previously defined notion of operator composition) using MultTable:

MultTable(sg)

4×4 MultTable{SymOperation{3}}:
──────┬────────────────────────
      │    1  2₀₀₁  2₀₁₀  2₁₀₀ 
──────┼────────────────────────
    1 │    1  2₀₀₁  2₀₁₀  2₁₀₀
 2₀₀₁ │ 2₀₀₁     1  2₁₀₀  2₀₁₀
 2₀₁₀ │ 2₀₁₀  2₁₀₀     1  2₀₀₁
 2₁₀₀ │ 2₁₀₀  2₀₁₀  2₀₀₁     1
──────┴────────────────────────

Alternatively, exploiting overloading of the *-operator, "raw" multiplication tables can be constructed via a simple outer product:

sg .* permutedims(sg) # equivalent to `reshape(kron(sg, sg), (length(sg), length(sg)))`

4×4 Matrix{SymOperation{3}}:
 1     2₀₀₁  2₀₁₀  2₁₀₀
 2₀₀₁  1     2₁₀₀  2₀₁₀
 2₀₁₀  2₁₀₀  1     2₀₀₁
 2₁₀₀  2₀₁₀  2₀₀₁  1

Symmorphic vs. nonsymorphic space groups

To determine whether a space group is symmorphic or not, use issymmorph taking either a SpaceGroup, LittleGroup, or SubperiodicGroup (or a SpaceGroup identified by its number and dimensionality; in this case, using tabulated look-up). To test whether a given SymOperation is symmorphic in a given centering setting, use issymmorph(::SymOperation, ::Char)

Group generators

Generators of SpaceGroups, PointGroups, and SubperiodicGroups are accessible via generators, e.g.:

ops = generators(sgnum, SpaceGroup{D})

2-element Vector{SymOperation{3}}:
 2₀₀₁
 2₀₁₀

To generate a group from a list of generators, we can use the generate method. As an example, we can verify that ops in fact returns symmetry operations identical to those in sg:

generate(ops)

GenericGroup{3}  with 4 operations:
 2₀₀₁
 2₀₁₀
 1
 2₁₀₀

Magnetic space groups

Magnetic space groups are accessible via mspacegroup.