In geometry, a polar point group is a

point group In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every ...

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 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 groups, 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 ...

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 Sucrose, a disaccharide, is a sugar composed of glucose and fructose subunits. It is produced naturally in plants and is the main constituent of white sugar. It has the molecular formula . For human consumption, sucrose is extracted and refi ...

, may exhibit triboluminescence.

References

{{reflist Symmetry Crystallography Group theory