Space group and crystal system

Going deeper

Symmetry, written as a code

A crystal is a pattern repeated in space, and it usually has more symmetry than the repetition alone: an axis it can be turned about, a plane it can be reflected in, a centre through which every atom has an identical partner on the opposite side. The space group collects all of these, together with operations that combine a turn or a reflection with a shift by part of a cell – a screw axis turns and advances, a glide plane reflects and slides. There are exactly 230 ways to combine them in three dimensions, and Volume A of the International Tables for Crystallography draws and tabulates every one.

The symbol reads from left to right. Its first letter gives the lattice – P for primitive, or a letter such as C, I, F or R for cells with extra lattice points – and the rest the symmetry along the principal directions. In P6₃/mmc, the space group of graphite, hBN and 2H-MoS₂, 6₃ is a sixfold screw axis along the stacking direction: turn by 60° and advance half a cell, which carries one layer onto the next. The /m is a mirror perpendicular to that axis, and the m and c that follow are mirror and glide planes along the other two sets of directions.

Seven crystal systems

Ignore the shifts and what is left is the point group, and the point groups sort crystals into seven crystal systems, each of which fixes the shape of the unit cell. Cubic has three equal edges at right angles; tetragonal two equal edges and a third different, all at right angles; orthorhombic three different edges at right angles; hexagonal and trigonal two equal edges at 120°, with the third at right angles to both; monoclinic one angle that is not a right angle; triclinic no constraint at all.

Most familiar 2D materials are hexagonal or trigonal: graphene, hBN and the common TMDCs. The orthorhombic ones – black phosphorus, SnSe, CrSBr – are those in which nothing in the symmetry makes the two in-plane directions equivalent, so mobility, optical absorption and magnetism can differ strongly along them. In monoclinic layered crystals such as CrI₃ at room temperature and α-RuCl₃, the axis that runs from one layer to the next leans away from the perpendicular, so each layer sits shifted sideways from the one below. Some crystals change system with temperature: bulk CrI₃ goes from monoclinic to rhombohedral stacking on cooling, and FeSe from tetragonal to orthorhombic near 90 K.

What the symmetry allows

A property of a crystal cannot be less symmetric than the crystal itself – Neumann’s principle – so the space group decides in advance what can and cannot be measured. A crystal with a centre of inversion can have no second-harmonic generation, no piezoelectricity and no spontaneous polarisation. That is why bulk 2H-MoS₂ produces no second harmonic while a single layer, which lacks inversion, produces a strong one, and why an effect the symmetry forbids, when it is seen, points to strain, defects, a substrate or a different phase.

Strictly, a single layer is not described by a space group at all, because it repeats in two directions only; it belongs to one of 80 layer groups. Databases usually report a monolayer in the space group of a computational cell – the layer repeated along the third direction with vacuum between the copies – which is why a data sheet can give one layer a symbol of its own. The bulk and monolayer groups differ whenever stacking adds or removes symmetry: thinning 2H-MoS₂ to one layer removes inversion, and thinning polar Td-WTe₂ to one layer restores it. The data sheets on this site give each material’s bulk space group from the Materials Project, and a monolayer’s from 2DMatPedia where it has one.