Polar point group

In geometry, a polar point group is a point group in which there is more than one point that every symmetry operation leaves unmoved.[1] The unmoved points will constitute a line, a plane, or all of space.

A straight line joining two unmoved points defines a unique axis of rotation, called a polar direction, unless the symmetry operations do not allow any rotation at all, such as mirror symmetry, in which case the polar direction must be parallel to any mirror planes.[clarification needed]

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 groupEdit

Of the 32 crystallographic point groups, 10 are polar:[2]

Polar crystallographic point groups
Crystal system Polar point groups
   Schönflies    Hermann–Mauguin Orbifold Coxeter
Triclinic C1 1 11 [ ]+
Monoclinic C2 Cs 2 m 22 * [2]+ [ ]
Orthorhombic C2v mm2 *22 [2]
Trigonal C3 C3v 3 3m 33 *33 [3]+ [3]
Tetragonal C4 C4v 4 4mm 44 *44 [4]+ [4]
Hexagonal C6 C6v 6 6mm 66 *66 [6]+ [6]
Cubic (none)

The space groups associated with a polar point group do not have a discrete set of possible origin points that are unambiguously determined by symmetry elements.[1]

When materials having a polar point group crystal structure are heated or cooled, they may temporarily generate a voltage called pyroelectricity.

Molecular crystals that occupy[clarification needed] polar space groups may exhibit triboluminescence.[3] A common example of this is sucrose, demonstrated by smashing a wintergreen lifesaver in a darkened room.


  1. ^ a b Jeremy Karl Cockcroft, Huub Driessen, David Moss, Ian Tickle (2006). "Polar Point Groups". University of London. Retrieved 2013-12-09.CS1 maint: uses authors parameter (link)
  2. ^ Kasap, Safa O. (2006). Principles of electronic materials and devices. Boston: McGraw-Hill. ISBN 9780073104645.
  3. ^ Zink, Jeffery (1981). "Triboluminescence-Structure Relations in Polymorphs of Hexaphenylcarbodiphosphorane and Anthranilic Acid, Molecular Crystals, and Salts". J. Am. Chem. Soc. 103: 1074–1079. doi:10.1021/ja00395a014.