Space group

SpaceGroup finds the space group of a crystal with spglib: its number and symbol, its symmetry operations, and the atoms they make equivalent. The same operations let phonons and elastic constants skip the displacements and strains that symmetry relates to others.

from mdapy.analysis import SpaceGroup
from mdapy.build import build_crystal

cell = build_crystal("l1_2", 3.75, elements=("Cu", "Au"))
result = SpaceGroup().compute(cell)
result.number, result.symbol  # 221, 'Pm-3m'
len(result.rotations)  # 48 operations
result.equivalent  # the Au corner on its own, the three Cu face centers together

spglib is an optional dependency: pip install spglib, or the extra symmetry of mdapy.

Parameters

Name	Type	Default	Unit	Meaning
tolerance	float64	1e-5	Å	atoms closer than it count as images of each other

compute(system) takes the cell of a crystal, periodic in every direction, of at most 2000 atoms: spglib searches its translations pair by pair, which takes minutes for 100,000 atoms where a unit cell takes milliseconds. Atoms of different elements are different; without an element column, atoms of different types; without either, all atoms are alike. Relaxed crystals keep their symmetry to well below the default tolerance; a structure whose atoms have moved more, such as a crystal at a temperature, needs a larger one to show the symmetry of its lattice.

Results

A SpaceGroupResult:

Attribute	Type	Meaning
number	int32	the number of the space group in the International Tables, 1 to 230
symbol	str	its Hermann–Mauguin symbol, such as "Fm-3m"
rotations	(K, 3, 3) float64	the rotations of the K symmetry operations, on Cartesian column vectors
translations	(K, 3) float64	their translations in Å: operation k moves the point r to rotations[k] @ r + translations[k]
equivalent	(N,) int64	the first atom that symmetry makes each atom equivalent to

The operations of a supercell include the translations of its lattice that are not translations of the supercell: a cubic cell of fcc has 192, four times the 48 of its primitive cell.

Errors

The constructor raises InvalidParameter for a tolerance that is not positive; compute raises it for a box that is not periodic, a system without atoms or of more than 2000, and without spglib.

Definition

The space group is the group of the rotations, reflections and translations that map the crystal onto itself, each atom onto an atom of its kind. spglib (Togo et al. 2024) finds it from the lattice and the positions, within the tolerance.