Polar point group

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In geometry, a polar point group is a point group in which every symmetry operation leaves more than one point unmoved.[1] Therefore, a point group with more than one axis of rotation or a mirror plane perpendicular to the axis of rotation cannot be polar.

A straight line joining unmoved points defines a unique axis of rotation, unless 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.

The space groups associated with a polar point group do not have their origins uniquely determined by symmetry elements.[1]

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]+ [ ]
Orthorombic C2v mm2 *22 [2]
Tetragonal C4 C4v 4 4mm 44 *44 [4]+ [4]
Trigonal C3 C3v 3 3m 33 *33 [3]+ [3]
Hexagonal C6 C6v 6 6mm 66 *66 [6]+ [6]
Cubic (none)

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


  1. ^ a b Jeremy Karl Cockcroft, Huub Driessen, David Moss, Ian Tickle (2006). "Polar Point Groups". University of London. Retrieved 2013-12-09. 
  2. ^ Kasap, Safa O. (2006). Principles of electronic materials and devices. Boston: McGraw-Hill. ISBN 9780073104645.