Lattice Canonical Form

CASM determines the canonical form of a lattice with respect to a point group, $g$, by (i) finding all equivalent spatial orientations of the Niggli cell of the lattice, (ii) applying all operations in $g$ to generate all symmetrically equivalent Niggli cell lattice vectors, and (iii) finding the lattice vectors (represented as a column vector matrix) that have the most standard orientation according to the following criteria:

• bisymmetric matrices are always more standard than symmetric matrices
• symmetric matrices are always more standard than non-symmetric matrices
• matrices with more positive values are preferred
• matrices with large values on the diagonal are preferred
• matrices with small off-diagonal values are preferred
• upper triangular matrices are preferred

For lattices without any basis, the appropriate point group is the lattice point group.

For supercells of a prim, the appropriate point group is the crystal point group.