Inversion is one of the fundamental symmetry operations of crystallography. The operation takes every point in the crystal and maps it through a single fixed point, the centre of inversion, to its diametrically opposite position. A crystal that contains a centre of inversion is called centrosymmetric; one that lacks such a centre is non-centrosymmetric. This single property has profound consequences for the optical and physical behaviour of the gem, particularly for the existence or absence of piezoelectricity and pyroelectricity.

Definition

If the centre of inversion is taken as the origin, then for every atom located at coordinates (x, y, z) there is an identical atom at (minus x, minus y, minus z). The operation is its own inverse, in that performing it twice returns every atom to its starting position. Inversion is one of the five basic operations from which all other point-group symmetries are built, alongside identity, rotation, mirror reflection and improper rotation (a rotation followed by a mirror).

Centrosymmetric and non-centrosymmetric structures

The 32 crystallographic point groups divide into 11 centrosymmetric and 21 non-centrosymmetric. The centrosymmetric ones, sometimes called Laue groups, share the property that all their physical tensors of even rank, such as elasticity, are unaffected by sign reversal. The non-centrosymmetric groups, by contrast, can support phenomena that depend on a directional asymmetry, including piezoelectricity (charge separation under mechanical stress) and second-harmonic generation in nonlinear optics. Of the 21 non-centrosymmetric groups, 20 support piezoelectricity, and a smaller subset of 10 supports pyroelectricity (charge generation under temperature change).

Implications for gem species

Quartz is the textbook example of a piezoelectric and non-centrosymmetric gem. Its trigonal point group 32 lacks a centre of inversion, which is why a quartz crystal develops surface charge under stress and is used in oscillators and pressure sensors. Tourmaline is both piezoelectric and pyroelectric, and a polished tourmaline crystal will attract dust and small particles to its analogous and antilogous poles when its temperature changes, a phenomenon noted in the literature since the eighteenth century. Diamond is centrosymmetric (point group m-3m) and shows neither piezoelectricity nor pyroelectricity, but its centrosymmetry contributes to the simplicity and high symmetry of its optical and elastic properties.

Inversion and optical activity

The absence of a centre of inversion is also the structural prerequisite for optical activity, the rotation of the plane of polarised light by a transparent material. Quartz is again the classical case: right-handed and left-handed quartz crystals rotate the plane in opposite senses, and the two enantiomorphs cannot be related by any pure rotation, only by inversion. The polariscope or rotating-polariser methods used in laboratory work to identify quartz exploit this directly.

Practical relevance

For most gem buyers, the language of inversion symmetry is invisible. For the gemmologist or laboratory analyst, however, it underpins the diagnostic procedures used to distinguish quartz, tourmaline and other piezoelectric species from their look-alikes, and it explains why certain stones develop surface charge in handling, attracting fibre and dust, while others do not.