Spacey.jl
The package provides functions related to finding the spacegroup of a crystal. (The spacegroup of a crystal is useful in the context density functional theory calculations, computational materials science, and machine learning for materials prediction.)
This package will eventually provide all of the functionality of the enumlib package and some additional functionality, useful for machine learning applications and materials science. In this pre-release version, only a pointgroup finder is provided, as well as a "snap to symmetry" function for unit cells.
One of the unique features of the package is its robustness to finite-precision issues, a constant bane of other spacegroup codes.
A few simple examples are provided below. This package finds symmetries very efficiently* by relying on the assumption that the basis is as compact as possible. To assure the most compact basis, this code first reduces the input basis vectors to a "Minkowski-reduced" basis. (See MinkowskiReduce.jl)
(* Normally efficiency is not important, since finding the symmmetryies is a relatively quick computation, but when the symmetries of tens of thousands of cases are needed in just a second or two, as in autoGR, efficiency becomes essential.)
Example 1: Hexagonal lattice
julia> u,v,w = ([1, 0, 0], [0.5, √3/2, 0.0], [0.0, 0.0, √(8/3)])
([1, 0, 0], [0.5, 0.8660254037844386, 0.0], [0.0, 0.0, 1.632993161855452])
julia> pointGroup(u,v,w)
24-element Vector{Matrix{Int64}}:
[-1 0 0; -1 1 0; 0 0 -1]
...
[1 0 0; 1 -1 0; 0 0 1]
By default the output is is "direct coordinates", the transformation matrices showing integer combinations of the basis that yield equivalent, but rotated basis vectors.
Example 2: Rhombohedral lattice, arbitrary rotation of coordinate system
julia> u = [1, 1, 2]; v = [1, 2, 1]; w = [2, 1, 1];
julia> u, v, w = threeDrotation(u, v, w, π / 3, π / 5, π / 7)
([1.0328466951537383, 1.855345731770484, 1.2210323173082058], [-0.2604424479658839, 2.2850084410500355, 0.8431525103014423], [0.7968127453125833, 2.3142904165695555, -0.09565206052008146])
julia> length(pointGroup(u, v, w)) == 12
true
julia> pointGroup(u, v, w)
12-element Vector{Matrix{Int64}}:
[-1 0 0; -1 1 1; 0 0 -1]
[-1 1 0; -1 0 -1; 0 0 1]
[-1 0 0; 0 -1 0; 0 0 -1]
[-1 1 0; 0 1 0; 0 0 1]
[0 -1 -1; -1 0 -1; 0 0 1]
[0 1 1; -1 1 1; 0 0 -1]
[0 -1 -1; 1 -1 -1; 0 0 1]
[0 1 1; 1 0 1; 0 0 -1]
[1 -1 0; 0 -1 0; 0 0 -1]
[1 0 0; 0 1 0; 0 0 1]
[1 -1 0; 1 0 1; 0 0 -1]
[1 0 0; 1 -1 -1; 0 0 1]
Example 3: Slightly distorted cubic case, "snap back" to perfect cubic cell
The lattice vectors are not quite orthogonal, not quite all the same length. First pointGroup_robust gets the symmetries that should be present for a perfect cubic cell. snapToSymmetry then finds the perfect cubic cell that is as close as possible to the original cell.
julia> u = [1+.01,0,0]; v = [0.,1-.01,0]; w = [0,0,1-.001];
julia> ops = pointGroup_robust(u,v,w)
48-element Vector{Matrix{Int64}}:
[-1 0 0; 0 -1 0; 0 0 -1]
[-1 0 0; 0 -1 0; 0 0 1]
[-1 0 0; 0 0 -1; 0 -1 0]
[-1 0 0; 0 0 1; 0 -1 0]
[-1 0 0; 0 0 -1; 0 1 0]
[-1 0 0; 0 0 1; 0 1 0]
[-1 0 0; 0 1 0; 0 0 -1]
[-1 0 0; 0 1 0; 0 0 1]
[0 -1 0; -1 0 0; 0 0 -1]
⋮
[1 0 0; 0 -1 0; 0 0 -1]
[1 0 0; 0 -1 0; 0 0 1]
[1 0 0; 0 0 -1; 0 -1 0]
[1 0 0; 0 0 1; 0 -1 0]
[1 0 0; 0 0 -1; 0 1 0]
[1 0 0; 0 0 1; 0 1 0]
[1 0 0; 0 1 0; 0 0 -1]
[1 0 0; 0 1 0; 0 0 1]
a,b,c,newops = snapToSymmetry(u,v,w,ops)
([0.9996332321644671, -1.1098158305294557e-16, 3.329447491588367e-16], [0.0, 0.9996332321644675, -3.330557677480862e-16], [0.0, -3.330557677480862e-16, 0.9996332321644676], [[-1 0 0; 0 -1 0; 0 0 -1], [-1 0 0; 0 -1 0; 0 0 1], [-1 0 0; 0 0 -1; 0 -1 0], [-1 0 0; 0 0 1; 0 -1 0], [-1 0 0; 0 0 -1; 0 1 0], [-1 0 0; 0 0 1; 0 1 0], [-1 0 0; 0 1 0; 0 0 -1], [-1 0 0; 0 1 0; 0 0 1], [0 -1 0; -1 0 0; 0 0 -1], [0 -1 0; -1 0 0; 0 0 1] … [0 1 0; 1 0 0; 0 0 -1], [0 1 0; 1 0 0; 0 0 1], [1 0 0; 0 -1 0; 0 0 -1], [1 0 0; 0 -1 0; 0 0 1], [1 0 0; 0 0 -1; 0 -1 0], [1 0 0; 0 0 1; 0 -1 0], [1 0 0; 0 0 -1; 0 1 0], [1 0 0; 0 0 1; 0 1 0], [1 0 0; 0 1 0; 0 0 -1], [1 0 0; 0 1 0; 0 0 1]])