The Necessary Conditions for Macroscopic Polarization in a Material

It is important to recognize that a useful piezoelectric effect is defined macroscopically. Each unit cell has to contribute constructively in order for the macroscopic effect to occur. It is the global symmetry that determines the macroscopic piezoelectric effect. For example, a piezoelectric ceramic containing randomly oriented crystal grains has no piezoelectric effect even though the symmetry of each unit cell allows piezoelectricity. A net polarization in the material is a sufficient but not a necessary condition for the presence of piezoelectricity for example, quartz is one of the popular piezocrystals without polarization. The existence of a polarization, however, does make the piezoelectric effect much more pronounced. In fact, the best piezoelectric materials are all ferroelectric materials. Most importantly, the hydrostatic piezoelectric effect belongs uniquely to polar materials. 

In a preceding paragraph we have shown that a necessary condition for second order NLO activity in a material is the absence of the symmetry center. This a stringent requirement and it is perhaps the most difficult feature the chemistry of these materials must face with. To get a non-centrosymmetric arrangement of molecules on a macroscopic scale is a difficult task, and even more difficult is to get a non-centrosymmetric polar structure. A polar structure is that in which there is at least one macroscopic direction (polar axis) which is not changed in the opposite direction by symmetry transformations allowed for the material. With reference to crystals, out of the 32 crystal classes only the following 20 correspond to non-centrosymmetric crystals 1,2, m, 222, mm2,4, -4,422,4mm, -42m, 3, 32, 3m, 6, —6, 622, 6mm, —62m, 23, —43m... 

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