Differences in atomic packing and bonding in different crystallographic directions influence a material’s mechanical, thermal, electrical, magnetic, and optical properties, all of which can be direction-dependent. The symmetry principles that govern the categorization of crystalline materials into seven crystal systems and 32 point groups also reflect in the symmetry of the material’s properties. This simple but powerful concept that the properties of a crystalline material must exhibit at least the same symmetry as the underlying crystal structure is Neumann’s principle.^S1

All crystalline substances, neglecting quasicrystals, belong to one of 32 crystallographic point groups. Eleven of those point groups possess an inversion center, which means that for any point (x,y,z) in the crystal, the corresponding point (-x,-y, -z) is completely identical. In other words, if you pick any point in space and turn the crystal “inside out” through that point, it will look and behave exactly the same as when you started. Of the remaining 21 point groups, 20 are piezoelectric (the combination of other symmetry operators prohibits piezoelectricity in point group 432), which simply means that applying a stress to the crystal will create a net separation of positive and negative charges in the crystal, hence the name piezoelectric, which translates simply to pressure-electric. This polarization results in the buildup of charges at the crystal surface. The converse is also true: applying an electric field to pull the positive and negative charges apart will produce a net strain in a piezoelectric material. It is this intrinsic coupling of electrical and mechanical energies that makes piezoelectricity such a useful phenomenon.