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Construct a parent lattice

Build a ParentLattice from a primitive basis matrix (Bravais) or a basis + dset (multilattice).

Setup

julia> using Enumlib

Bravais lattice (single-site)

Pass the basis matrix alone. Enumlib uses column-major basis matrices: column j of A is the j-th lattice vector in Cartesian coordinates.

julia> A_fcc = [0.0 0.5 0.5;
                0.5 0.0 0.5;
                0.5 0.5 0.0];

julia> p = ParentLattice(A_fcc); # If not specified, one atomic site in the dset, at (0,0,0) 

julia> ndset(p)                  # one dset position, at the origin
1

julia> length(space_group(p))    # 48 ops (Pm-3m point group of FCC)
48

Multilattice (HCP, diamond, perovskite, …)

Pass the basis plus a dset of basis-site fractional positions:

julia> A_hcp = [1.0 -0.5 0.0;
                0.0 sqrt(3)/2 0.0;
                0.0 0.0 sqrt(8/3)];

julia> dset_hcp = [[0.0, 0.0, 0.0], [1/3, 2/3, 1/2]];

julia> p_hcp = ParentLattice(A_hcp, dset_hcp);

julia> ndset(p_hcp) # The dset has two atomic sites
2

julia> n_nonzero_translations(p_hcp)   # P6_3/mmc has 12 screw ops of 24
12

What gets canonicalized silently

  1. Each dset position is adjusted to be in between 0 and 1: wrapped into [0,1)^D via mod(·, 1).
  2. For lattices with only one atomic site ("Bravais" lattices), the lone dset entry is automatically shifted to the origin so the space group doesn't carry "artificial" translations from the user's choice of origin.
  3. Bases with Hadamard ratio |det(A)| / prod(‖aⱼ‖) ≤ 1e-12 are rejected as flat, not defining a lattice in 3D. The Hadamard ration is a scale-free singularity check that works whether you express positions in Ångströms, nm, Bohr, or anything else.

See also