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] 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.

Polar crystallographic point group

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

Polar crystallographic point groups
Crystal system	Polar point groups
Crystal system	Schönflies		Hermann–Mauguin		Orbifold		Coxeter
Triclinic	C₁		1		11		[ ]⁺
Monoclinic	C₂	C_s	2	m	22	*	[2]⁺	[ ]
Orthorhombic	C_2v		mm2			*22		[2]
Trigonal	C₃	C_3v	3	3m	33	*33	[3]⁺	[3]
Tetragonal	C₄	C_4v	4	4mm	44	*44	[4]⁺	[4]
Hexagonal	C₆	C_6v	6	6mm	66	*66	[6]⁺	[6]
Cubic	(none)

The space groups associated with a polar point group do not have their origins uniquely 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 polar space groups will exhibit triboluminescence.^[3] A common example of this is sucrose, demonstrated by smashing a wintergreen lifesaver in a darkened room.

References

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

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[uol-1] 1 2 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.

[tribo-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.