In geometry, a polar point group is a point group in which there is more than one point that every

symmetry operation In group theory, geometry, representation theory and molecular symmetry, a symmetry operation is a transformation of an object that leaves an object looking the same after it has been carried out. For example, as transformations of an object in spac ...

leaves unmoved. The unmoved points will constitute a line, a plane, or all of space. While the simplest point group, C₁, leaves all points invariant, most polar point groups will move some, but not all points. To describe the points which are unmoved by the symmetry operations of the point group, we draw a straight line joining two unmoved points. This line is called a polar direction. The

electric polarization In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...

must be parallel to a polar direction. In polar point groups of high symmetry, the polar direction can be a unique axis of rotation, but if the symmetry operations do not allow any rotation at all, such as mirror symmetry, there can be an infinite number of such axes: in that case the only restriction on the polar direction is that it must be parallel to any mirror planes. A point group with more than one axis of rotation or with a mirror plane perpendicular to an axis of rotation cannot be polar.

Polar crystallographic point group

Of the 32

crystallographic point group In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation (perhaps followed by a translation) would leave the structure of a crystal un ...

s, 10 are polar: The

space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unch ...

s associated with a polar point group do not have a discrete set of possible origin points that are unambiguously determined by symmetry elements. When materials having a polar point group crystal structure are heated or cooled, they may temporarily generate a voltage called pyroelectricity. Molecular crystals which have symmetry described by one of the polar space groups, such as sucrose, may exhibit

triboluminescence Triboluminescence is a phenomenon in which light is generated when a material is mechanically pulled apart, ripped, scratched, crushed, or rubbed (see tribology). The phenomenon is not fully understood, but appears to be caused by the separation ...

References

{{reflist Symmetry Crystallography Group theory