A crystal has inversion symmetry if turning it inside out through a point – replacing every atom by the one directly opposite – leaves it looking the same. Many of the most useful effects in appear only when that symmetry is missing.
Going deeper
Inversion maps every atom at r onto an identical atom at −r. A honeycomb of two different atoms has no such centre, and in 2H-MoS2 the symmetry comes and goes with the layer count – which second-harmonic light shows directly, bright for odd layers and nearly dark for even ones.
What the symmetry forbids
A crystal has inversion symmetry if there is a point through which every atom at position r has an identical partner at −r. Any property that would change sign under that operation must then vanish. That rules out a spontaneous electric polarisation, and – for electric-dipole processes – , all of which need a direction that inversion would reverse.
Inversion also shapes the bands. Together with it keeps every band doubly -degenerate at every momentum. When inversion is broken, can split the bands by spin: this is the origin of at interfaces and of the large spin splitting of the at K in .
Set by layer count and stacking
In layered materials, whether a sample has an inversion centre depends on how many layers it has and how they are stacked, not only on its chemistry. Graphene has one; a monolayer of hBN does not. In -MoS2 and its relatives, odd numbers of layers lack inversion and even numbers have it, so piezoelectricity and second-harmonic generation switch on and off layer by layer. Rhombohedral 3R stacking lacks inversion at every thickness, and its second-harmonic signal grows as layers are added.
Stacking can also be engineered. Two hBN or TMDC layers placed parallel rather than antiparallel break inversion and carry a small out-of-plane polarisation that flips when one layer slides over the other – .
How it is checked
Second-harmonic generation is the quickest test: a sample without inversion converts part of an intense laser beam to twice its frequency, and rotating the polarisation reveals the crystal’s orientation. force microscopy detects the mechanical response to an electric field. and infrared spectra help as well: in a centrosymmetric crystal, no vibration is both Raman- and infrared-active.
Ordinary is a poor judge, because diffraction patterns look centrosymmetric even when the crystal is not. Local effects also blur the answer: edges, , and applied fields break inversion locally, and even-layer samples still give a weak signal from surfaces and higher-order processes. “No signal” in practice means much weaker, not exactly zero.
For specialists
Invariance under r → −r about a centre. Its absence is the precondition for piezoelectricity, a switchable polarisation, second-harmonic generation, Rashba splitting and -contrasting optical selection rules, and it decides whether Weyl or Dirac nodes are allowed. In layered crystals it follows layer number and stacking rather than chemistry alone: a 2H TMDC monolayer lacks it, the bilayer has it, and sliding one layer over another removes it again.