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Why can only crystal structures without a centre of symmetry be piezoelectric?

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Figure 1: Ferroelectric properties of barium titanate (BaTiO3). (Left) Above 120° C the structure of the BaTiO3 crystal is cubic, and there is no net polarization of charge; (right) below 120° C the structure changes to tetragonal, shifting the relative positions of the ions and causing a concentration of positive and negative charges toward opposite ends of the crystal. — capacitor dielectric and piezoelectric ceramics Figure 1: Ferroelectric properties of barium titanate (BaTiO₃). (Left) Above 120° C the structure of the BaTiO₃ crystal is cubic, and there is no net polarization of charge; (right) below 120° C the structure changes to tetragonal, shifting the relative positions of the ions and causing a concentration of positive and negative charges toward opposite ends of the crystal.

Piezoelectricity occurs due to the relative displacement of ions within a crystal lattice when the material is subjected to mechanical stress.12

For a material to exhibit this effect, its crystal structure must lack a centre of symmetry—also known as an inversion centre. A centre of symmetry is a point within the crystal from which the structure is identical in any two opposite directions. If a crystal possesses this symmetry, the displacement of ions caused by stress in one direction is effectively cancelled out by the symmetry of the lattice, preventing the development of a net electric polarization.21

In materials like barium titanate, this symmetry is lost during a structural transition (such as from cubic to tetragonal), which shifts ions away from their central positions and allows for the generation of a voltage under stress.2