sugar). The dipole density or polarization (dimensionality [C·m/m3] ) may easily be calculated for crystals by summing up the dipole moments per volume of the crystallographic unit cell. As every dipole is a vector, the dipole density P is a vector field. Dipoles near each other tend to be aligned in regions called Weiss domains. The domains are usually randomly oriented, but can be aligned using the process of poling (not the same as magnetic poling), a process by which a strong electric field is applied across the material, usually at elevated temperatures. Not all piezoelectric materials can be poled. Of decisive importance for the piezoelectric effect is the change of polarization P when applying a mechanical stress. This might either be caused by a reconfiguration of the dipole-inducing surrounding or by re-orientation of molecular dipole moments under the influence of the external stress. Piezoelectricity may then manifest in a variation of the polarization strength, its direction or both, with the details depending on: 1. the orientation of P within the crystal; 2. crystal symmetry; and 3. the applied mechanical stress. The change in P appears as a variation of surface charge density upon the crystal faces, i.e. as a variation of the electric field extending between the faces caused by a change in dipole density in the bulk. For example, a 1 cm3 cube of quartz with 2 kN (500 lbf) of correctly applied force can produce a voltage of 12500 V. Piezoelectric materials also show the opposite effect, called the converse piezoelectric effect, where the application of an electrical field creates mechanical deformation in the crystal. Mathematical description Linear piezoelectricity is the combined effect of The linear electrical behavior of the material: where D is the electric flux density (electric displacement), ε is the permittivity (free-body dielectric constant), E is the electric field strength, and , . Hooke's law for linear elastic materials: where S is the linearized strain, s is compliance under short-circuit conditions, T is stress, and , where u is the displacement vector. These may be combined into so-called coupled equations, of which the strain-charge form is: where is the piezoelectric tensor and the superscript t stands for its transpose. Due to the symmetry of , . In matrix form, where [d] is the matrix for the direct piezoelectric effect and [d] is the matrix for the converse piezoelectric effect. The superscript E indicates a